ii} 


STATISTICAL  AVERAGES 


A  Methodological  Study 


BY 


DR.  FRANZ  ZIZEK 

Privat-Docent  in  Statistics  in  the  University  of  Vienna 


AUTHORIZED  TRANSLATION 

WITH  ADDITIONAL  NOTES   AND  REFERENCES 

BY 

WARREN  MILTON  PERSONS 

Professor  of  Economics  in  Colorado  College 


NEW   YORK 

HENRY  HOLT  AND  COMPANY 

1913 


COPTTSIGHT,   1913, 
BY 

HENRY  HOLT  AND  COMPANY 


THE    OUINN   A    BODEN    CO.  PRESS 
RAHWAY,    N,  J. 


g  2^9 -5  E 


% 

£ 

< 

I  TRANSLATOR'S  PREFACE 

3 

*'3      I  have  long  felt  the  Avant  of  a  general  methodological 
work  to  be  used  as  the  basis  of  a  college  course  in  statistics. 
Die  statistischen  Mittelwerte,  by  Dr.  Franz  Zizek,  seemed 
to  me   to  meet  the  requirements   of  a  non-mathematical 
text-book  on  statistics  better  than  any  work  available  in 
English  and,   consequently,  this  English  translation  was 
undertaken. 
?        Statistical  Averages,  when  used  as  a  text-book,  should, 
'*^    of   course,    be   supplemented   by   lectures,   assigned  read- 
S    ings,    and     statistical    problems.     Lectures    and    assigned 
readings  should  cover  such  topics  as  the  history  of  statistics, 
the  organization  and  work  of  labor  and  census  bureaus, 
the  preparation  of  schedules,  the  accuracy  obtainable  in 
statistics,  the  construction  and  use  of  diagrams  and  maps, 
and  the  mathematical  methods  of  computing  the  standard 
deviation   and   the   coefficient   of   correlation.     Numerous 
problems  should  be  assigned  so  that  the  students  may  be- 
come familiar  vnth  frequency  tables,  graphic  representa- 
tion, the  methods  of  computing  the  important  index  num- 
bers (such  as  the  Labor  Bureau  index  numbers  of  wages 
and  prices  and  the  Economist  index  number  of  prices),  the 
standard   deviation,    correlation   tables,   the   coefficient  of 
correlation,  and  the  like.    By  the  assignment  of  problems 
the   students   will   become    familiar  with   the   sources   of 
statistical" data  and  with  the  facts  of  statistics  as  well  as 
with  the  methods  of  statistics. 
J         Perhaps    the    most    important    advantage    that    Statis- 
\     tical  Averages  offers  to  American  readers  is  to  be  found  in 
)     the  explanations  of,  and  copious  references  to,  statistical 
:     data    and    methods    from    French,    German,    and   Italian 


iv  TRANSLATOR'S  PREFACE 

sources.  Probably  the  chief  defect  of  the  book  is  the  lack 
of  illustrative  matter  in  the  way  of  graphic  representation 
and  statistical  tables  of  various  sorts.  However,  this  de- 
fect can  be  turned  to  advantage  in  class-room  use  by  re- 
quiring the  students  to  secure  such  matter  and  present  it, 
graphically  and  otherwise,  as  indicated  in  the  preceding 
paragraph. 

A  few  additions  have  been  made  to  the  translation  in  the 
way  of  footnotes  signed  "  Translator,"  new  titles  added  to 
the  bibliography,  and  selected  references  to  those  statistical 
journals  and  publications  which  are  of  especial  use  to 
American  readers. 

In  conclusion,  I  acknowledge  my  obligation  to  Dr.  Franz 
Zizek,  who  cheerfully  authorized  the  translation  and  who 
corrected  the  manuscript,  to  Mr.  J.  C.  Schwartz,  who  aided 
in  the  translation,  to  Professor  T.  S.  Adams  of  the  Univer- 
sity of  Wisconsin,  who  read  the  proof,  and  especially  to 
Professor  W.  K.  Stewart  of  the  German  Department  of 
Dartmouth  College,  who  prepared  a  considerable  portion  of 
the  translation  and  who  aided  in  other  ways. 

W.  M.  P. 


CONTENTS 


PAGE 

Introduction 1 

PART  I 

STATISTICAL  AVERAGES  IN  GENERAL 

CHAPTER 

I.     CLASSIFICATION  OF  STATISTICAL  SERIES  WITH 

REFERENCE  TO  THE  PROBLEM  OF  AVERAGES   .        7 

II.  ISOLATED  AVERAGES  AND  STATISTICAL  EVO- 
LUTION FROM  ISOLATED  AVERAGES  TO  AVER- 
AGES BASED  UPON  SERIES  OF  ITEMS      ...      25 

A.  Isolated  Averages  Obtained  by  Computation    .      25 

B.  Estimated  Isolated  Averages        ....      33 

(a)  Estimation  of  the  average  size  of  a  virtual  individ- 
ual element  of  observation         34 

(b)  Estimation  of  relative  numbers  which  are  them- 
selves averages 39 

C.  Some  Instances  op  the  Progress  op  Statistics 

AWAY  FROM  ISOLATED  AVERAGES  AND  TOWARD  AV- 
ERAGES Based  on  Series  op  Items  (Average 
Length  op  Life.  Length  op  Marriage,  Length 
op  a  Generation,  Number  op  Children  per 
Family) 41 

III.  NECESSITY  OF  THE  LOGICAL  AGREEMENT  OF 
MAGNITUDES  FROM  WHICH  AN  AVERAGE  IS  TO 

BE  COMPUTED 60 

IV.  POSTULATE  OP  THE  GREATEST  POSSIBLE  HOMO- 
GENEITY OF  SERIES  FROM  WHICH  AVERAGES 
ARE  COMPUTED.  AND  OF  MASSES  WHICH  ARE 
CHARACTERIZED  BY  RELATIVE  NUMBERS    .        .      65 

A.  Postulate  op  the  Greatest  Possible  Homoge- 
neity op  Single  Observations  from  Which  an 
Average  Is  Computed 66 

B.  Postulate  op  the  Greatest  Possible  Homogene- 
ity in  the  Computation  op  an   Average  from 


vi  CONTENTS 

CHAPTER  PASB 

Values  Wnicn  Express  the  Sizes  op  Masses 
Limited  in  a  Definite  Way  (Constituents  of 
A  Greater  Totality) 72 

C.  Postulate  op  the  Greatest  Possible  Homoge- 
neity OF  ]\Iasses  Which  Are  Characterized  by 
Relative  Numbers 73 

D.  Simple  Indication  of  the  Range  op  a  Series     .      77 

V.     FORxMATION  OF  MAGNITUDE  CLASSES    ...  80 

VI.     NATURE  AND  PURPOSE  OF  AVERAGES   ...  93 

A.  The  Computation  op  Averages  for  Their  Own 
Sake 93 

B.  Averages  for  Purposes  op  Comparison       .        .  93 

1.  General  reasons  for  the  application  of  averages  for 
purposes  of  comparison 93 

2.  Mean  index  numbers 95 

3.  Meaning  of  the  postulate  of  the  greatest  possible 
homogeneity  for  the  comparison  of  averages  and 
relative  numbers 101 

(a)  Postulate  of  t1ie  greatest  possible  Jiomogeneity  in 
the  comparison  of  averages  obtained  from  series 

of  individual  obsei'vations  ....     102 

(b)  Postulate  of  tJie  greatest  possible  homogeneity  in 
tlie  co)nparison  of  averages  from  series  whose 
members  indicate  the  size  of  masses  limited  in  a 
definite  way 106 

(c)  Postulate  of  the  greatest  possible  homogeneity  in 

tlie  comjMrison  of  relative  numbers  .        .        .     106 

4.  Investigation  of  causation  by  comparison  of  aver- 
ages and  relative  numbers 110 

C.  Averages  as  Standards  for  Judging  Items    .        .    122 

D.  The  Function  op  Averages  in  the  Measurement 

op  the  Dispersion  op  Series 125 


PART  II 

THE  VARIOUS  KINDS  OF  AVERAGES 

I.    SYNOPSIS 131 

II.    THE  ARITHMETIC  MEAN 138 


CONTENTS  vii 

CHAPTER  PAGE 

A.  The    Simple    Arithmetic    Mean,    or,  Shortly, 
Arithmetic  Mean 138 

1.  Concept  aud  qualities  of  the  arithmetic  mean       .  138 
3.  Distinction  between  statistical  series  with  refer- 
ence to  the  computation  of  the  arithmetic  mean  .  139 

3.  Computation  of  the  arithmetic  mean    .        .        .  143 

4.  Application  of  the  arithmetic  mean       .        .        .  148 

B.  The  Weighted  Arithmetic  Mean    .        .        .        .150 

C.  The  Arithmetic  Mean  and   ]\Iathematical  Sta- 
tistics    163 

1.  The  arithmetic  mean  of  series  of  quantitative 
individual  observations  and  the  theory  of  errors 
of  observation 164 

3.  The  arithmetic  mean  of  series  of  statistical  proba- 
bilities and  the  theory  of  probability     .        .         .     171 

3.  Relation  of  "mathematical  "  to  "  non-mathemati- 
cal "  statistics  ;  typical  aud  atypical  arithmetic 
means 178 

4.  Mathematical  methods  of  judging  the  signifi- 
cance of  the  difference  between  two  arithmetic 
means  or  statistical  probabilities  ....     186 

III.  THE  GEOMETRIC  MEAN 194 

IV.  THE  MEDIAN 199 

1.  Concept  and  properties  of  the  median  .        .        .  199 

3.  Series  in  which  the  median  can  be  determined      .  803 

3.  Determination  of  the  median        ....  305 

4.  Application  of  the  median 315 

V.     THE  MODE 333 

1.  Concept  and  properties  of  the  mode     .        .        .  833 

8.  Determination  of  the  mode 333 

3.  Applications  of  the  mode 840 

PART  III 

DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

I.    PURPOSE  AND  J^IEANING  OP  ESTABLISHING  THE 

DISPERSION  OF  STATISTICAL  SERIES     .        .        .351 


vi  CONTENTS 

CHAPTER  PABB 

Values  Wnicn  Express  the  Sizes  op  Masses 
Limited  in  a  Definite  Way  (Constituents  op 
A  Greater  Totality) 73 

C.  Postulate  op  the  Greatest  Possible  Homoge- 
neity OF  ]\Iasses  Which  Are  Characterized  by 
Relative  Numbers 73 

D.  Simple  Indication  of  the  Range  op  a  Series     .      77 

V.     FORxAIATION  OF  MAGNITUDE  CLASSES    ...  80 

VI.     NATURE  AND  PURPOSE  OP  AVERAGES   ...  93 

A.  The  Computation  op  Averages  for  Their  Own 
Sake 93 

B.  Averages  for  Purposes  op  Comparison       .        .  93 

1.  General  reasons  for  the  application  of  averages  for 

purposes  of  comparison 93 

8.  Mean  index  numbers 95 

3.  Meaning  of  the  postulate  of  the  greatest  possible 
homogeneity  for  the  comparison  of  averages  and 
relative  numbers 101 

(a)  Postulate  of  tlie  greatest  possible  liomogeneity  in 
the  comparison  of  averages  obtained  from  series 

of  individual  observations         ....     103 

(b)  Postulate  of  ilie  greatest  possible  homogeneity  in 
tJie  com'parison  of  averages  from  series  whose 
members  indicate  the  size  of  masses  limited  in  a 
definite  way 106 

(c)  Postulate  of  the  greatest  possible  homogeneity  in 

the  comparison  of  relative  numbers  .        .        .     106 

4.  Investigation  of  causation  by  comparison  of  aver- 
ages and  relative  numbers 110 

C.  Averages  as  Standards  for  Judging  Items    .        .    138 

D.  The  Fonction  op  Averages  in  the  Measurement 

op  the  Dispersion  of  Series 135 


PART  II 
THE  VARIOUS  KINDS  OF  AVERAGES 

I.    SYNOPSIS 131 

II.    THE  ARITHMETIC  MEAN 138 


CONTENTS  vii 

CHAPTER  PAGE 

A.  The    Simple    Arithmetic    Mean,   or,  Shortly, 
Arithmetic  Mean 138 

1.  Concept  and  qualities  of  the  arithmetic  mean       .  138 

2.  Distinction  between  statistical  series  with  refer- 
ence to  the  computation  of  the  arithmetic  mean  .  139 

3.  Computation  of  the  arithmetic  mean    .        .        .  142 

4.  Application  of  the  arithmetic  mean       .        .        .  148 

B.  The  "Weighted  Arithmetic  Mean    .        .        .        .150 

C.  The  Arithmetic  Mean  and   ]\Iathematical  Sta- 
tistics    163 

1.  The  arithmetic  mean  of  series  of  quantitative 
individual  observations  and  tlie  theory  of  errors 

of  observation 164 

2.  The  arithmetic  mean  of  series  of  statistical  proba- 
bilities and  the  theory  of  probability     .        .         .     171 

3.  Relation  of  "mathematical  "  to  "  non-mathemati- 
cal "  statistics  ;  typical  and  atypical  arithmetic 
means 178 

4.  Mathematical  methods  of  judging  the  signifi- 
cance of  the  difference  between  two  arithmetic 
means  or  statistical  probabilities  ....     186 

III.  THE  GEOMETRIC  MEAN 194 

IV.  THE  j^IEDIAN 199 

1.  Concept  and  properties  of  the  median  .        .        .  199 

2.  Series  in  which  the  median  can  be  determined      .  202 

3.  Determination  of  the  median        ....  205 

4.  Application  of  the  median 215 

V.    THE  MODE 222 

1.  Concept  and  properties  of  the  mode     .        .        .  222 

2.  Determination  of  the  mode 233 

3.  Applications  of  the  mode 240 

PART  III 

DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

I.    PURPOSE  AND  IMEANING  OF  ESTABLISHING  THE 

DISPERSION  OF  STATISTICAL  SERIES     .        .        .251 


viii  CONTENTS 

CHAPTER  PAGK 

II.     THE  DISPERSION  OF  SERIES  OF  QUANTITATIVE 

INDIVIDUAL  OBSERVATIONS 256 

A.  Measurement  and  Presentation  op  the  Disper- 
sion OF  Series  op  Quantitative  Individual  Ob- 
servations BY  Means  of  Elementary  Mathemat- 
ical methods 256 

B.  Measurement  op  the  Dispersion  op  Series  op 
Quantitative  Individual  Observations  prom  the 
Standpoint  of  the  Theory  op  Error   .        .        .     270 

III.  THE  DISPERSION  OF  SERIES  COMPOSED  OP  MAG- 

NITUDES LIMITED  IN  A  DEFINITE  WAY  (CON- 
STITUENTS OF  A  GREATER  TOTALITY)  .        .     292 

IV.  THE  DISPERSION  OF  SERIES  OF  RELATIVE  NUM- 

BERS AND  AVERAGES  WHICH  CHARACTERIZE 
MASSES  LIMITED  IN  A  DEFINITE  WAY  (CON- 
STITUENTS OF  A  GREATER  TOTALITY)  IN  SOME 
OTHER  RESPECT  THAN  AS  REGARDS  THEIR 
MAGNITUDES 297 

A.  The  General  Problem  (Distinguishing  Time, 
Space,  and  Qualitative  or  Quantitative  Series)    297 

B.  Elementary  Mathematical  Methods  for  Meas- 
uring AND  Representing  the  Dispersion  of  Series 
OF  Relative  Numbers  and  Averages  Which 
Characterize  Masses  Limited  in  a  Definite  Way 
(Constituents  of  a  Greater  Totality)  in  Some 
Other  Respect  than  According  to  Magnitude  .     316 

C.  Examination  op  the  Dispersion  op  Series  op 
Numerical  Probabilities  by  the  Methods  op 
THE  Theory  op  Probability 319 


APPENDIX  I 

A.  Series  Which  Exhibit  a  Characteristic  Regu- 
lar Distribution  op  Items  in  Other  Ways  than 
WITH  Respect  to  the  Dispersion  op  the  Items 
About  Their  Mean  (Series  of  Characteristic 
Conformation) 341 

B.  Investigation  of  Causes  upon  the  Basis  op 
Quantitative  Series  of  Characteristic  Confor- 
mation   352 


CONTENTS  ix 

PAGE 

C.  Investigation   of   Causes  Through   Comparison 

OP  Geographic  and  Time  Series     ....  359 

D.  Correlation  Between  Individual  Characters   .  370 

APPENDIX  II 

Quetelet's  "Average  Man" 373 

APPENDIX  III 

Bibliography 380 

Index 385 


STATISTICAL  AVERAGES 

INTRODUCTION 

In  nearly  all  branches  of  statistical  investigation  averages 
or  means  (French,  "  moyennes,"  German,  "  Mittelwerte, " 
Italian,  "  medie  ")  are  very  often  used  for  various  pur- 
poses of  the  greatest  significance.  Averages  are,  therefore, 
undoubtedly  to  be  reckoned  among  the  most  important  aids 
ir  statistical  method.  Even  Guerry,  the  founder  of  moral 
statistics,  indicated  the  extraordinary  importance  of  aver- 
ages in  the  following  definition  of  statistics:  "  The  science 
of  statistics  consists  essentially  in  the  methodological  enu- 
meration of  variable  elements,  whose  mean  it  determines." 
Edgeworth  defines  statistics  as  "  the  science  of  those 
means  which  are  presented  by  social  phenomena  "^;  and 
similarly  Bowley  says,  "  Statistics  may  rightly  be  called 
the  science  of  averages."^  The  application  of  averages 
has,  as  is  well  known,  given  rise  to  controversies  of  various 
kinds,  which  fill  a  considerable  part  of  statistical  literature. 
Not  infrequently  the  incorrect  use  of  averages  has  also  led 
to  erroneous  conclusions  and  to  contradictions,  which  have 
shaken  our  confidence  in  statistics.  Averages,  indeed,  are 
only  applicable  under  strictly  defined  conditions,  and  con- 
clusions based  on  averages  are  likewise  permissible  only 
within  well  defined  limits.  It  is  the  task  of  statistical 
science  to  investigate  the  application  and  use  of  averages 
from  the  general  methodological  standpoint,  and  to  deter- 
mine the  part  which  averages  should  play  in  statistical 
method. 

^ "  On  Methods  of  Statistics,"  Jubilee  Volume  of  the  Royal  Statis- 
tical Society   (1885),  p.  182. 

=  Elements  of  Statistics,  2nd  ed.   (1902),  p.  7. 


2  INTRODUCTION 

Statistical  literature  does,  indeed,  possess  a  large  number 
of  works  which  deal  with  isolated  questions  connected  with 
the  application  of  averages  or  with  the  various  averages  in 
use  in  definite  departments  of  applied  statistics  (for  ex- 
ample, the  average  length  of  life,  or  average  number  of 
children  per  family).  Especially  the  "  mathematical  statis- 
ticians "  (Lexis,  Edgeworth,  Westergaard,  von  Bortkiewicz, 
Pearson,  Galton,  Yule,  Bowley,  and  others),  as  well  as 
some  philosophers  and  theoretical  mathematicians,  who  have 
also  occupied  themselves  with  statistical  problems  (as,  for 
example,  Fechner,  J.  von  Kries,  Czuber,  and  Blaschke), 
have  thoroughly  investigated  various  methodological  ques- 
tions connected  with  averages  by  using  the  calculus  of 
probabilities.^  But  only  a  few  works  have  for  their  object 
the  treatment  of  statistical  averages  in  the  most  general 
methodological  manner.*  As  a  matter  of  fact,  not  one 
of  the  treatises  referred  to  offers  anything  like  an  exhaust- 
tive  development  of  the  problem.  Moreover,  most  of  these 
works  are  decidedly  out  of  date. 

Furthermore,  the  discussions  regarding  averages,  which 
are  to  be  found  in  the  numerous  handbooks,  text-books,  and 

•  The  mathematical  investigations  of  formal  theories  of  popula- 
tion and  the  measurement  of  mortality  by  such  authors  as  Becker, 
G.  F.  Knapp,  Zeuner,  and  Wittstein  have  little  bearing  on  our 
problem. 

*  Among  such   independent  studies   are: 

Bertillon,  Adolphe,  "  La  theorie  des  moyennes  en  Statistique," 
Journal  de  la  Society  de  Statistique  de  Paris,  1876. 

Bertillon,  Adolphe,  article  entitled  "  Moyenne "  in  the  Diction- 
naire  encyclopedique  des  sciences  m^dicales. 

Edgeworth,  F.  Y.,  article  entitled  "  Average "  in  Palgrave's  Dic- 
tionary of  Political  Economy. 

Fechner,  G.  Th.,  Kollektivmasslehre,  published  by  G.  F.  Lipps, 
Leipzig,   1897. 

Holmes,  George  K.,  "A  Plea  for  the  Average,"  Quar.  Pubs,  of  the 
Am.  Stat.  Assoc,  New  Series  No.  16,  December,  1891. 

Messedaglia,  Angelo,  "  II  calcolo  dei  valori  medl  e  le  sue  appli- 


INTRODUCTION  3 

systematic  treatises,  as  well  as  in  the  investigations  devoted 
to  statistical  method  in  general,  are  for  the  most  part  rather 
meager.  General  works  on  statistics  and  statistical  method 
cannot,  in  the  nature  of  things,  give  much  space  to  the  in- 
dividual problems  in  which  statistics  abound.^ 

In  the  following  pages  the  attempt  will  be  made  to  offer 
a  systematic  treatment  of  the  most  important  questions 
which  make  up  the  problem  of  statistical  averages.  The 
scope  of  these  questions  is  extremely  wide,  as  a  very  great 
number  of  statistical  methods  depend  upon  the  application 
of  averages,  and  as  the  most  important  objects  of  statistical 
investigation  demand  the  use  of  averages.  For  these 
reasons  the  problem  of  averages  may  be  correctly  designated 
as  one  of  the  most  important  in  scientific  statistics,  and  in 
a  sense  as  the  central  problem  of  statistics.  Since  statistics 
by  its  nature  deals  with  phenomena,  both  variable  and  com- 
plicated, it  is  evident  that  averages,  which  characterize 
such  phenomena  by  a  single  number,  must  be  of  preeminent 
importance  in  the  science. 

Our  aim  in  the  following  treatment  is  a  general  methodo- 
logical one.  That  is  to  say,  our  aim  is  to  determine  those 
properties  which  the  various  types  of  averages,  such  as, 
arithmetic  mean,  geometric  mean,  median,  mode,  etc.,  pos- 
sess intrinsically,  irrespective  of  the  department  of  statis- 
tics (population  statistics,  economic,  moral,  biological  sta- 

cazioni  statist! che,"  Arch,  di  Stat.,  Anno  V,  1880.  The  same  in 
French :  "  Caleul  des  valeurs  moyennes,"  Annales  de  demographic 
Internationale,  IV,  1880. 

Quetelet,  A.,  "  Sur  I'appreciation  des  moyennes,"  Bull.  Commiss. 
Central.  Statist.,  Vol.  II,  1845. 

Quetelet,  A.,  Lettres  sur  la  theorie  des  probabiIit€s,  Pt.  II,  "  Des 
moyennes  et  des  limites,"  1845. 

Tammeo,  G.,  Le  medie  e  loro  limiti,   1878. 

Venn,  J.,  "  On  the  Nature  and  Use  of  Averages,"  Jour,  of  the 
Roy.  Stat.  Soe.,  1891. 

"  See  Appendix  III  of  this  book  for  a  list  of  titles  of  general 
works  on  statistics. 


4  INTRODUCTION 

tistics,  etc.)  to  which  the  numerical  data,  from  which  the 
average  is  computed,  belong.  We  shall  show  that  the  same 
methodological  problems  constantly  recur  in  the  most  di- 
verse fields  of  statistics,  and  that  these  problems  have  a 
common  solution,  although  heretofore  each  field  has  been 
worked  independently  without  aid  from  the  results  ob- 
tained in  other  fields.  It  is  also  important  to  note  that 
this  method  of  handling  averages  is  in  accordance  with 
the  most  significant  evolutionary  tendency  of  modern  sta- 
tistics. 

The  author  does  not  possess  sufficient  mathematical  train- 
ing to  be  able  to  apply  or  examine  critically  the  methods 
of  * '  mathematical  statistics, ' '  which — apart  from  the  theory 
of  the  development  and  growth  of  population — are  all  con- 
nected with  the  problem  of  averages.  Nevertheless,  such 
methods  will  not  be  entirely  disregarded.  On  the  contrary, 
in  order  to  supplement  the  exposition  of  the  elementary 
mathematical  methods,  it  will  be  shown  upon  what  funda- 
mental principles  tne  methods  of  '  *  mathematical  statistics  ' ' 
depend,  to  what  problems  these  methods  have  been  applied, 
and  what  interesting  results  have  been  attained.  The 
author  deems  some  consideration  of  "  mathematical  sta- 
tistics "  indispensable  because  its  problems  do  not  differ 
essentially  from  those  of  elementary  scientific  statistics. 
By  the  application  of  the  calculus  of  probability  the  * '  math- 
ematical statistician  "  attempts,  with  mathematical  pre- 
cision, to  solve  problems  which  confront  any  scientifically 
minded  investigator.  Since  the  majority  of  statisticians 
are  trained  in  economics  and  not  in  mathematics  the  author 
will  attempt  to  give  in  non-technical  language  some  infor- 
mation of  the  processes  and  results  of  mathematical  in- 
vestigations in  statistics.  In  this  way  the  significance  of 
the  calculus  of  probability  in  general  statistical  method  wiU 
become  apparent. 


PART  I 
STATISTICAL  AVERAGES  IN  GENERAL 


CHAPTER  I 

CLASSIFICATION    OF     STATISTICAL     SERIES     WITH 
REFERENCE  TO  THE  PROBLEM  OF  AVERAGES 

Statistical  series  are  classified  in  various  ways  in  the  text- 
books of  statistics.  They  are  usually  differentiated  as  space, 
time,  and  qualitative  or  quantitative  series,  according  as 
the  individual  members  of  the  series  are  distinguished  by 
their  distribution  in  space  (geographical  divisions)  or  time, 
or  by  their  qualitative  or  quantitative  differences.  It  is 
also  customary  to  classify  statistical  series  according  as 
their  individual  members  are  absolute  numbers,  or  relative 
numbers  or  averages.  With  reference  to  the  problem  of 
averages  a  classification  of  a  particular  kind  is  appropriate. 
The  various  series  are,  therefore,  embraced  in  three  groups 
according  to  the  different  methods  of  ascertaining  the  aver- 
ages. These  three  groups  must  also  be  differentiated  in 
the  discussion  of  the  different  special  problems. 

In  the  first  group  are  to  be  included  those  series  of  ob- 
servations upon  individuals  or  units  of  various  kinds  which, 
for  the  purpose  in  mind,  are  treated  as  similar.  In  these 
series  each  individual  member  refers  to  an  observa- 
tion unit  which  is  marked  by  some  defined  character.  This 
character  may  be  qualitative  or  quantitative.  We  have  to 
deal  "wath  a  qualitative  character  when,  for  example,  the 
sex  or  the  occupation  of  certain  individuals  is  being  con- 
sidered. 

Since,  however,  qualitative  individual  observations  do 
not  permit  the  computation  of  an  average  they  do  not 
enter  further  into  our  problem.  For  that  reason  we  shall 
deal  exclusively  in  the  following  pages  with  quantitative  in- 

7 


8  STATISTICAL  AVERAGES  IN  GENERAL 

dividual  observations.  Such  observations  arise  ordinarily 
through  measurement.  Thus,  for  example,  the  age,  vv^ages, 
income,  length  of  life,  etc.,  of  single  individuals  in  definite 
groups  of  the  population  are  measured  and  are  then  repre- 
sented in  the  form  of  series.  Cases  occur,  however,  in 
which  the  items  contained  in  the  series  do  not  deal  with 
real  measurements  but  with  quantitative  observations  of  an- 
other kind,  such  as  are  obtained  by  counting.  Thus,  for  in- 
stance, houses  are  observed  with  reference  to  the  number  of 
their  occupants,  families  with  reference  to  the  number  of 
children.® 

From  series  of  quantitative  observations  various  kinds  of 
averages  may  be  computed,  of  which  the  most  important 
are  the  arithmetic  mean,  the  median  (that  is,  the  middle 
number  of  the  series  when  the  items  are  arranged  accord- 

'  Series  of  quantitative  individual  observations,  whether  they  be 
actual  measurements  or  quantitative  individual  observations  of  an- 
other kind,  are  either  space,  time,  qualitative,  or  quantitative  series. 
The  units  to  which  the  observations  refer  frequently  belong  to  dif- 
ferent space  or  time  divisions;  thus,  for  example,  the  domiciles  of 
persons  whose  age,  income,  etc.,  are  measured  present  space  differ- 
ences, and  the  data  giving  ages  at  marriage  or  death  present  time 
differences.  But  the  space  or  time  differentiation  of  the  individual 
observations,  which  appears  in  the  original  material,  normally  dis- 
appears during  the  course  of  the  statistical  work  and  is  not  evident 
in  the  resulting  statistical  series.  Series  of  quantitative  individual 
observations  are,  furthermore,  not  qualitative  series,  since  similar 
units  are  selected  for  measurement.  Neither  are  series  of  quantita- 
tive individual  observations  identical  with  the  group  of  quantitative 
series — as  many  authors  appear  to  assume — because  only  those 
series  are  to  be  designated  as  "  quantitative "  whose  items  are  dif- 
ferentiated from  one  another  by  some  quantitative  criterion,  as,  for 
example,  is  the  ca.se  with  series  of  death  rates,  birth  rates,  etc.,  for 
different  age  classes  of  the  population.  The  fact  that  quantitative 
individual  observations  possess  different  numerical  values  for  the 
element  of  observation  does  not  constitute  them  quantitative  series, 
since  all  series  (time,  space,  etc.)  consist  of  numbers  of  various 
sizes.  The  above-mentioned  customary  division  of  statistical  series 
into  space,  time,  qualitative,  and  quantitative  groups  is,  therefore, 
not  exhaustive. 


CLASSIFICATION  OF  STATISTICAL  SERIES         9 

ing  to  size,  or,  in  case  there  is  an  even  number  of  items, 
the  arithmetic  mean  of  the  two  middle  numbers),  and  the 
mode  (that  is,  the  relatively  most  frequent  value,  the  point 
of  greatest  density).  The  average  computed  from  the 
series  represents  the  mean  of  the  measurement  in  question 
(average  age,  average  wage,  average  income,  mean  and 
probable  lifetime)  ;  or  else  it  indicates,  when  a  definite 
quantitative  character  is  obtained  by  counting,  how  many 
units  of  that  character  occur  on  the  average  (for  instance, 
the  average  number  of  occupants  per  house,  or  children 
per  family). 

Quantitative  observations  are  not  always  presented  with 
the  greatest  possible  detail.  Often  the  variant  items  are 
tabulated  according  to  class  (for  example,  age,  income, 
wage  classes,  etc.).  The  frequency  table,  thus  obtained, 
merely  indicates  (absolutely  or  relatively)  how  many  items 
belong  to  the  different  classes.  Averages  may  be  com- 
puted from  the  frequency  tables  for  the  character  in  ques- 
tion, either  measured  or  counted,  no  matter  whether  the 
tables  consist  of  absolute  or  relative  numbers. 

The  items  which  produce  the  series  in  question  may 
belong  to  the  most  varied  branches  of  social  life.  More- 
over, similar  series  may  arise  from  observations  in  natural 
science.  Especially  meteorological  (thermal  and  baromet- 
ric) observations,  and  also  anthropological  measurements 
(height,  chest-girth,  various  dimensions  of  the  skull,  mus- 
cular power,  lung  capacity,  etc.),  produce  series  which  are 
well  adapted  to  the  application  of  statistical  methods. 
Likewise,  series  of  measurements  of  certain  characters  of 
animals  and  plants  have  recently  been  investigated  accord- 
ing to  the  methods  of  scientific  statistics.  Indeed,  the  most 
important  methods  of  modern  mathematical  statistics  have 
been  developed  from  biological  material,  and  statistical 
method  plays  as  important  a  part  in  modern  biology  as 
it  does  in  sociology.  In  particular,  the  questions  of  varia- 
tion and  heredity  are  being  investigated  with  great  success 


10  STATISTICAL  AVERAGES  IN  GENERAL 

by  use  of  the  statistical  method.  Instead  of  speaking  of 
individuals  we  may,  therefore,  speak,  with  Gustav  Theo- 
dor  Fechner,  of  ''collective  objects."'^  Fechner  under- 
stands by  a  collective  object  one  which  consists  of 
an  indefinite  number  of  accidentally  varying  speci- 
mens, which  are  grouped  by  a  generic  notion.  Man, 
in  general,  according  to  Fechner,  forms  a  collective  object 
in  the  wider  sense;  man  of  a  definite  sex  or  age  forms  a 
collective  object  in  the  narrower  sense.  Meteorological  ob- 
servations, anthropometric  data,  measurements  of  animals 
and  plans,  etc.,  represent  other  collective  objects  whose 
chance  variation  the  science  of  collective  masses  ("  Kollek- 
tivmasslehre  ")    investigates. 

All  the  illustrations  quoted  have  had  to  do  with  series 
of  observations  which  refer  to  different  individuals  with 
some  common  characteristic.  The  items  of  a  series,  how- 
ever, may  arise  from  repeated  observations,  especially 
measurements,  of  the  same  object.  The  mean  computed 
from  such  series  is  called  "  objective,"  in  contradistinction 
to  the  "  subjective  "  mean  computed  from  series  of  single 
observations  of  a  number  of  units.® 

Statistics,  as  a  social  science,  deals  constantly  with  series 
of  single  observations  of  various  similar  units  and,  there- 
fore, with  "  subjective  "  means  in  the  above  sense.  We 
take  the  wages,  the  income,  the  age  of  various  individuals 
and  compute  the  average.  Likewise,  the  means  computed 
from  meteorological  observations,  anthropometric  data,  and 

'  Cf.  Kollektivmasslehre  by  G.  T.  Fechner.  Lipps  and  Bruns  may 
also  be  mentioned  as  supporters  of  "  Kollektivmasslehre." 

*  Cf.  A.  Bertillon,  "  La  th6orie  des  moyennes  en  Statistique "  in 
the  Journal  de  la  Soci6t6  de  Statistique  de  Paris,  17th  year,  p.  266; 
also  J.  Bertillon,  Cours  6l6mentaire  de  Statistique  administrative, 
p.  112,  and  G.  v.  Mayr,  Theoretische  Statistik,  p.  98.  Block 
(Trait6  th^orique  et  pratique  de  Statistique,  2nd  ed.,  p.  129)  re- 
jects the  classification  of  averages  into  "  objective  "  and  "  subjective  " 
means  for  the  unsatisfactory  reason  that  it  is  necessary  to  reserve 
the  notion  of  a  "  moyenne  typique  "  for  the  means  "  qu'on  prend  sur 


CLASSIFICATION  OF  STATISTICAL  SERIES        11 

biological  measurements  are  "  subjective."  On  the  other 
hand,  in  other  fields,  especially  astronomy  and  geodesy,  re- 
peated measurements  are  often  made  of  the  same  object, 
in  which  cases  an  "  objective  "  mean  is  computed.  For 
example,  the  problem  may  be  to  determine  the  declination 
of  a  star  or  the  zenith-distance  of  its  path  across  a  definite 
meridian,  or  else  to  measure  the  latitude  of  a  place  or  the 
distance  between  two  points.  By  computing  the  arithmetic 
mean  of  repeated  measurements  the  "  most  probable  "  size 
of  the  object  in  question  is  obtained.  This  average,  in  all 
probability,  gives  most  nearly  the  true,  or  ideal,  size  of  the 
object  whose  measurements  are  affected  by  accidental 
errors. 

It  was  with  reference  to  such  "  objective  "  means  that 
the  principles  of  the  theory  of  errors  were  developed,  espe- 
cially by  Gauss.  These  were  first  applied  by  Quetelet  to 
series  of  anthropometric  data,  and  later,  by  mathematical 
statisticians,  to  other  series  of  individual  observations. 
Such  studies  have  shown  that  the  characteristic  distribution 
of  the  items  about  their  mean,  which  marks  the  series  of 
measurements  of  the  same  object  (called  "  normal  "  dis- 
tribution), holds  sometimes  also  for  series  of  meas- 
urements of  similar  units.  Where  this  is  the  case,  the 
similar  items  may  be  regarded  as  empirical  values,  affected 
by  accidental  errors,  and  in  such  cases  the  average,  which 
represents  the  ideal  value,  gains  additional  significance. 
This  average   may  be   regarded   as  the   resultant  of  the 

une  s^rie  de  mesures  op6r6es  sur  un  meme  objet,  ou  qui  ne  s'ap- 
pliquent  qu'a  des  grandeurs  peu  differentes."  Block  believes,  then, 
that  means  of  series  of  repeated  measurements  of  the  same  object  and 
means  of  series  of  small  dispersion  may  be  classed  together  as 
"  typical "  means.  But  the  distinction  between  series  according  as 
they  originate  from  repeated  measurements  of  the  same  object  or 
single  measurements  of  different  objects  should  not  be  confused  with 
the  distinction  between  series  according  to  the  kind  of  their  disper- 
sion. The  notion  of  a  "  typical "  mean  should  be  reserved  for  means 
of  series  with  a  definite  dispersion. 


12  STATISTICAL  AVERAGES  IN  GENERAL 

complex  of  general  causes  producing  the  single  items, 
and  is  a  "  typical  "  mean  in  the  strictest  sense  of  the 
word. 

Thus  the  mathematical  theory  developed  from  repeated 
measurements  of  the  same  object  affords  a  special  basis 
for  the  arithmetic  mean  of  certain  series  of  single  measure- 
ments of  various  similar  units  and  gives  us  a  standard  for 
judging  the  dispersions  of  such  series.  It  is  to  be  noted, 
however,  that  the  Gaussian  law  of  normal  distribution  holds 
for  series  of  measurements  of  different  units  only  in  isolated 
cases.  Anthropometric  series  of  this  kind  have  sometimes 
been  established,  especially  measurements  of  height.  Lexis 
has  proved  that  the  dispersion  of  items  about  the  "  nor- 
mal ' '  length  of  life  is  in  accordance  with  the  theory  of  er- 
rors. On  the  other  hand,  Fechner  and,  especially,  Pearson 
have  noted  numerous  statistical  series  which  do  not  corre- 
spond to  the  Gaussian  law,  but  which,  in  spite  of  their  asym- 
metry, may  be  brought  under  a  generalized  law  of  accidental 
variation.  In  general,  then,  it  appears  that  the  normal  dis- 
tribution holds  for  repeated  observations  of  the  same  object, 
but  that  the  series  of  observations  of  different  objects,  as 
a  rule,  show  an  unsymmetrical  distribution  about  the  aver- 
age. 

A  further  distinction  between  series  of  measurements  of 
the  same  object  and  of  similar  objects  may  be  noted. 
In  series  of  repeated  measurements  of  the  same  object  we 
are  concerned  with  establishing  the  true  size  of  the  object 
with  the  greatest  possible  accuracy.  Such  series  follow,  as 
has  been  said,  the  Gaussian  law,  and  the  three  means,  the 
arithmetic  mean,  the  median,  and  mode,  theoretically  co- 
incide. In  series  whose  members  refer  to  different,  even 
though  similar,  objects,  it  is  not  proper  to  speak  of  a 
**  true  "  value.  All  measurements  of  such  objects,  however 
much  they  may  differ  from  their  mean,  are  equally  true.  It 
is,  therefore,  simply  a  question  of  applying  the  most  suitable 
and  comprehensive-  descriptive  term  to  the  group  of  meas- 


CLASSIFICATION  OF  STATISTICAL  SERIES        13 

urements.  For  this  purpose  the  statement  of  the  average  or 
averages  is  indispensable.  If  the  series  follows  the  Gauss- 
ian law,  then,  even  in  the  case  of  single  measurements  of 
different  units  the  three  means,  above  mentioned,  coincide. 
If,  on  the  contrary,  the  series  does  not  follow  this  law 
then  the  three  means  diverge  from  one  another  and  should, 
if  possible,  all  be  stated,  since  each  of  them  contributes 
something  unique  towards  characterizing  the  series. 

In  consequence  of  this  essential  distinction  between  ob- 
jective and  subjective  means,  a  different  importance  is  to 
be  attached  to  the  dispersion  about  the  means  of  the  two 
series.  The  dispersion  of  a  series  of  measurements  of  the 
same  object  depends  exclusively  upon  the  accuracy  of  the 
measuring  instruments  and  determines  the  degree  of  pre- 
cision of  the  mean.  The  magnitude  that,  added  to  the 
arithmetic  mean,  and  subtracted  from  such  mean,  gives 
two  limits  between  which  a  given  proportion  of  the  measure- 
ments fall,  may  be  taken  as  an  index  varying  inversely  with 
precision  of  the  instrument.^  The  dispersion  of  series  of 
measurements  of  various  units  plays  a  more  substantial 
role,  since  each  item  of  such  a  series  is  to  be  regarded  as  a 
fact  determined  by  peculiar  causes.  The  dispersion  of  the 
series,  therefore,  indicates  the  variability  of  the  phenome- 
non in  question. 

Consequently,  the  distinction  between  subjective  and  ob- 
jective means  is  of  importance  in  several  directions. 
If  we  conceive  statistics  merely  as  a  social  science,  then 
it  follows  that  the  "  objective  "  mean  plays  no  part  in  it. 
If  we,  however,  investigate  the  theoretical  foundation  of 
the  statistical  method,  then  it  is  necessary  to  become  famil- 
iar with  the  mathematical  principles  which  have  developed 
from  consideration  of  repeated  observations  of  the  same 
object,  and  which  are  now  also  applied  to  the  "  subjective  " 
mean.  In  the  latter  case  we  must,  of  course,  investigate 
the  modifications  required  in  applying  these  principles  to 

•  Cf.  G.  Th.  Fechner,  Kollektivmasslehre,  p.  15. 


14  STATISTICAL  AVERAGES  IN  GENERAL 

series  of  measurements  of  similar  objects  taken  from  the 
fields  of  sociology,  biology,  meteorology,  etc. 

A  second  group  which  we  wish  to  differentiate  may  be 
illustrated  by  the  population  figures  for  the  various  dis- 
tricts of  a  country.  The  members  of  such  series  give  the 
size  of  definitely  limited  groups,  aggregates,  or  masses 
(e.g.,  counties),  which,  taken  together,  form  a  totality  of 
a  higher  order  (e.g.,  state).  Stated  in  general  terms, 
the  second  group  embraces  those  series  whose  members 
give  the  size  of  definitely  limited  masses  ^*  which,  taken 
together,  form  a  totality  of  a  higher  order.  The  members 
of  such  series  are  not,  like  those  of  the  first  group,  similar 
items,  either  measurements  or  other  elements  of  observa- 
tion, but  they  are  statistical  masses,  mutually  limited  as 
to  size  by  the  point  of  view  taken  in  the  particular  problem. 
The  definition  of  the  masses  may  be  from  the  point  of  view 
of  space,  time,  qualitative,  or  quantitative  differences.  The 
average  computed  from  the  series  gives  the  mean  value  of 
the  masses  belonging  to  it,  that  is,  the  average  number  of 
units  per  mass. 

If,  for  instance,  we  posses  the  population  figures  for  the 
various  districts  of  a  country  (space  masses),  we  may  then 
determine  the  average  population  per  district.  If  we  have 
the  number  of  births  or  deaths  for  a  series  of  years  (time 
masses),  we  may  reckon  how  many  births  or  deaths  occur 

'a  The  word  "  mass  "  has  been  used  throughout  this  translation  to 
designate  a  group  of  units.  The  science  of  statistics,  of  course, 
passes  from  consideration  of  groups  of  concrete  units  to  detailed 
examination  and  analyses  of  abstract  figures  characterizing  such 
groups.  Thus,  the  science  deals  directly  with  series  of  items.  How- 
ever, the  results  obtained  from  examination  of  abstract  figures  or 
items  are  made  useful  only  by  connecting  them  with  some  concrete 
mass.  In  this  way  statistics  becomes  an  applied  science  of  masses. 
The  use  of  the  word  "  mass  "  in  this  translation  accords  with  its 
use  in  the  name  which  has  been  given  to  the  mathematical  instru- 
ment which  has  been  invented  for  rendering  statistical  data  available 
for  scientific  purposes,  i.  e.,  "the  calculus  of  mass  phenomena"  (see, 
for  instance,  H.  L.  Moore's  Laws  of  Wages,  p.  4). — Teanslatoe. 


CLASSIFICATION  OF  STATISTICAL  SERIES        15 

on  an  average  per  year  of  the  given  period.  If  we  have 
data  as  to  the  sizes  of  the  different  occupations  (qualita- 
tive masses)  or  of  age  classes  of  the  population  (quantita- 
tive masses),  we  may  find  out  the  average  number  of 
persons  per  occupation  or  age  class.  But  such  averages 
computed  from  qualitative  and  quantitative  series  are,  as 
a  rule,  worthless,  since  the  average  size  of  a  mass  depends 
solely  on  the  number  of  masses,  the  statistician  himself 
generally  determining  arbitrarily  the  number  of  qualitative 
or  quantitative  masses  into  which  he  divides  the  totality  of 
the  higher  order.  For  example,  the  greater  the  number  of 
occupation  or  age  classes,  the  fewer  will  be  the  number  of 
people  assignable  to  a  single  one.^'*  On  the  other  hand,  when 
the  statistician  deals  with  space  or  time,  he  is  handling 
objective  magnitudes  (years,  districts,  etc.),  and  the  com- 
putation of  the  average  size  of  a  mass  in  such  a  sense  may 
be  of  great  significance. 

From  the  series  of  the  second  group,  as  a  rule,  only  the 
arithmetic  mean  is  taken.  The  prerequisites  for  the  appli- 
cation of  the  other  means  do  not  normally  occur.^^ 

'"  For  similar  reasons  it  is  useless  to  compute  the  average  size 
of  a  constituent  from  a  series  of  percentages  which  gives  the  space 
or  time  sizes  of  the  constituents  in  relation  to  a  totality  of  a  higher 
order.  The  size  of  tlie  average  of  such  a  series  depends  solely  upon 
the  number  of  constituents.  For  example,  suppose  we  have  statistical 
data  for  ten  years  or  classified  for  ten  sections  of  a  country,  or  if  we 
divide  the  population  into  ten  occupation  groups  or  into  ten  age 
classes,  then  one  year,  one  section,  one  occupation  group  or  one  age 
class  would,  of  course,  contain,  on  the  average,  10^  of  the  cases. 

^'  The  quantitative  series  of  the  second  group  are,  considered  from 
another  point  of  view,  at  the  same  time  members  of  the  first  group, 
that  is,  they  are  series  of  individual  observations  which  are  em- 
braced by  the  quantitative  constituents  (magnitude  classes).  A 
series,  for  example,  which  gives  the  distribution  of  the  population 
according  to  age  class  consists  of  individual  age  data  combined  into 
magnitude  classes.  The  same  holds  true  for  series  showing  the 
number  of  persons  receiving  various  classes  of  income  or  wages. 
If  we  consider  such  series  as  series  of  the  second  group  and  com- 


16  STATISTICAL  AVERAGES  IN  GENERAL 

A  third  group  consists  of  those  series  whose  members  are 
not  absolute  but  relative  numbers.  These  relative  num- 
bers may  be  either  subordinate  or  coordinate.  Sub- 
ordinate numbers  are  those  which  indicate  the  relative 
size  of  parts  to  a  whole,  usually  by  percentages;  for  ex- 
ample, the  percentage  of  male  and  of  female  births  to  the 
total  number  of  births,  or  the  percentage-distribution  of 
deaths  according  to  sex,  age,  or  social  class.  Coordinate 
numbers  are  those  which  indicate  the  relative  size  of  pairs 
of  coordinate  masses.  They  originate  through  the  interre- 
lation of  two  such  masses,  namely,  by  dividing  one  by  the 
other  and  sometimes  multiplying  the  quotient  by  100  or 
1,000.  Thus  the  coordinate  number,  obtained  by  multi- 
plying the  ratio  of  the  number  of  deaths  to  the  number 
of  living  by  1,000,  indicates  the  death  rate  per  thousand. 
In  the  same  way,  we  may  obtain  the  marriage  rate  or 
birth  rate  per  thousand  of  population.  Neither  subordinate 
nor  coordinate  numbers  arise  from  simple  measurement 
or  counting,  but  always  from  computation.^^* 

The  masses,  which  are  characterized  by  the  relative  num- 
bers, may  have  originated  from  a  criterion  of  space,  time, 
quality,  or  quantity.  Thus  we  may,  for  instance,  represent 
the  sex  distribution  of  infants  by  a  series  of  subordinate 
numbers  for  various  districts,  months,  religious  denomina- 
tions, or  age  classes  of  the  parents  according  as  we  use  a 
criterion  of  space,  time,  quality,  or  quantity  in  dividing 

pute  the  mean  size  of  a  quantitative  constituent,  as  explained  above, 
we  obtain  averages  of  less  significance  than  if  we  consider  them  to 
be  series  of  the  first  group  and  compute  the  mean  size  of  the  element 
of  observation  (age,  income,  wage,  etc.),  which  is  of  the  highest 
significance. 

i»aWhen  relative  numbers  are  stated  in  a  numerical  form  such 
that  if  used  to  multiply  a  total  (e.  g.,  population)  an  allied  num- 
ber (e.  g.,  number  of  births)  will  result,  then  such  relative  numbers 
are  called  "statistical  coefficients."  "Thus,  if  the  birth-rate  is 
40  per  1,000,  the  coefficient  is  .04"  (Bowley,  Elements  of  Statistics, 
p.  129). — Translator. 


CLASSIFICATION  OF  STATISTICAL  SERIES        17 

the  total  number  of  births  in  a  year,  and  compute  the  sex- 
ratio  for  each  of  these  divisions.  Similarly  we  may  secure 
death  rates  for  various  districts,  months  of  the  year,  re- 
ligious denominations,  or  age  classes  by  taking  the  ratios 
of  the  number  of  deaths  to  the  correlated  population,  both 
deaths  and  population  being  differentiated  according  to  a 
criterion  of  space,  time,  quality,  or  quantity.  These  ratios 
form  a  series  of  coordinate  numbers. 

It  is  characteristic  of  the  third  group  of  series  that  the 
items  generally  refer  to  masses  of  varying  size  and  therefore 
of  varying  significance  or  weight.  But  the  relative  im- 
portance is  not  apparent  from  the  items  themselves.  It 
is  of  the  nature  of  relative  numbers  that  they  do  not 
express  the  absolute  size  of  the  masses  from  which  they 
originate;  1,000  men  and  1,000  women  give  the  same  sex 
ratio  as  1,000,000  of  each,  and  50  deaths  among  2,000 
living  give  the  same  ratio  as  2,500  deaths  among  100,000 
living. 

Whether  the  series  be  composed  of  subordinate  or  of 
coordinate  numbers,  whether  the  criterion  be  that  of  space, 
time,  quality,  or  quantity,  it  is  evident  in  general  that 
items  arising  from  different  districts,  different  years,  differ- 
ent occupations  or  different  age  classes  must  produce  rela- 
tive numbers  of  varying  weights.  Because  of  the  vary- 
ing weights,  a  mean  should  not  usually  be  computed 
directly  from  items  of  the  third  group,  as  it  was  in 
the  cases  of  the  first  and  second  groups.  On  the  con- 
trary, the  mean  of  such  series  is  independent  of  the  indi- 
vidual items,  and  it  must  be  computed  from  the  original 
data  which  gave  rise  to  the  items  of  the  series.  Thus,  in 
order  to  get  the  true  average  annual  death  rate  for  a  period 
of  10  years,  the  total  number  of  deaths  during  such  period 
must  be  divided  by  the  sum  of  those  living  at  the  begin- 
ning (or  other  definite  point  of  time)  of  each  of  the  10 
years. 

The  mean  is  therefore  not  to  be  computed  from  the  series 


18  STATISTICAL  AVERAGES  IN  GENERAL 

but,  found  independently,  it  supplements  the  items  of  the 
series.  The  original  data  are  used,  then,  in  whole  or 
in  part,  to  compute  the  individual  items  of  a  series  and 
their  mean,  both  items  and  mean  being  relative  numbers. 
In  point  of  time  the  computation  of  the  items  may  sometimes 
follow  the  computation  of  the  average  value  for  the  totality. 
In  fact  the  tendency  of  statistical  research  has  been  as  a 
first  step  to  ascertain  general  averages  and  secondly  to 
separate  the  statistical  data  into  more  homogeneous  parts, 
for  which  separate  relative  numbers  are  computed.  Thus, 
general  death  rates  are  followed  by  death  rates  for  various 
age  classes,  occupations,  etc. 

Since  the  mean  and  the  items  are  computed  indepen- 
dently, it  may  also  happen  that  not  all  of  the  components 
of  the  total  are  taken.  For  example,  the  probability  of 
death  may  be  computed  for  the  whole  population  and  for 
certain  easily  defined  occupations,  while  other  occupations 
are  disregarded. 

From  what  has  been  said,  it  follows  immediately  that 
the  computation  of  the  different  means,  possible  for  the 
first  group,  is  not  possible  for  the  third.  The  general 
average  originates  from  the  absolute  numbers  found  for 
the  totality.  These  absolute  numbers  are  the  sum  of  the 
component  absolute  numbers,  from  which  the  items  were 
computed.  The  general  average  is  thus  really  obtained  by 
weighting  the  individual  items,  and  consequently  it  may 
be  computed  directly  from  such  items  by  finding  the 
weighted  arithmetic  mean.  Consequently,  if  the  original 
data  are  not  available,  we  may  compute  a  weighted  arith- 
metic average  from  the  items  of  the  series,  assigning  weights 
as  nearly  correct  as  possible.  If  we  only  know 
that  the  respective  weights  are  not  essentially  dif- 
ferent (such  as  may  occur  in  a  time  series  pertain- 
ing to  a  population  which  does  not  change  essentially 
during  the  period  considered),  we  may  be  contented  with 
a  simple  arithmetic  average  of  the  items  of  the  series.     In 


CLASSIFICATION  OF  STATISTICAL  SERIES        19 

case  the  weights  are  identical  the  simple  arithmetic  mean 
of  the  items  will  coincide  with  the  general  average  com- 
puted independently  from  the  original  data. 

In  anthropometry  several  relative  numbers  are  used  (for 
example,  the  ratio  of  the  length  of  the  head  to  its  breadth, 
or  the  cephalic  index)  which  differ  from  the  relative  num- 
bers computed  from  demographic  data  in  that  they  origi- 
nate through  correlating,  not  masses,  but  single  measure- 
ments. Averages  are  frequently  computed  from  such  rela- 
tive numbers ;  for  example,  average  cephalic  index.  As  the 
dividends  and  divisors,  which  give  the  ratios,  differ  among 
themselves  the  average  computed  directly  from  the  various 
ratios  is  different  from  the  quotient  of  the  sum  of  the 
dividends  and  of  the  divisors.  An  illustration  may  be 
taken  from  Dr.  Bertillon's  **  Theorie  des  moyennes."  ^^  If 
we  measure  two  skulls — limiting  ourselves  to  two  for  sake  of 
simplicity — and  find  the  first  200  mm.  long  by  180  mm. 
broad,  and  the  second  160  mm.  long  by  112  mm.  broad,  the 
cephalic  indices  will  be  90  and  70,  respectively;  giving  an 
average  index  directly  computed  of  80.  But  dividing 
the  sum  of  200  and  160  by  180  plus  112  we  obtain  the 
average  81.2.  Mathematicians  and  anthropologists  differ 
as  to  which  method  is  more  correct.  The  results  obtained 
by  the  two  methods  differ,  as  a  rule,  but  little.^^ 

Those  relative  numbers  have  a  particular  significance 
which  are  expressed  as  numerical  probabilities,  or  known 
functions  of  them.  Lexis  defines  a  statistical  prob- 
ability as  a  fraction  whose  numerator  gives  a 
number  of  observed  special  eases  or  elements,  which 
either  origiimte  from  the  number  of  observed  cases 
or  elements   given   in   the   denominator,    or  are   actually 

"  Journ.  de  la  Soc.  de  Stat,  de  Paris,  1876,  p.  314. 

"  Cf.  Chap.  XXII  of  Fechner's  Kollekth-masslehre:  "  Kollektive 
Behandlung  von  Verhaltnissen  zwischen  Dimensionen ;  mittlere  Ver- 
baltnisse"    (§§147-151,  pp.  352-364). 


20  STATISTICAL  AVERAGES  IN  GENERAL 

included  in  such  denominator.^*  He  calls  the  former 
case  of  probability  relation  genetic  or  primary,  the  latter, 
analytic  or  secondary.  * '  In  the  former  case  the  numerator 
gives  the  number  of  events  of  a  particular  kind  which 
originate  from  the  totality  forming  the  denominator;  for 
instance,  the  ratio  of  death  in  a  definite  age  class  to  the 
number  living  who  have  reached  the  lower  limit  of  the 
class  in  question  and  have  been  subject  to  the  death  risk 
in  question."  ^^  "In  the  case  of  analytic  probability  rela- 
tion, on  the  contrary,  the  units  of  the  numerator  are  of 
the  same  kind  as  those  of  the  denominator  and  are  only 
distinguished  by  some  special  characteristic ;  the  numerator, 
therefore,  forms  a  section  of  the  totality  expressed  by  the 
denominator.  Such  is  the  ratio  of  the  number  of  male 
births  to  the  total  number  of  births. ' '  ^®  Genetic  numerical 
probabilities  (for  instance,  the  probability  of  death)  are,  in 
our  terminology,  coordinate  numbers;  analytic  numerical 
probabilities  correspond  to  subordinate  numbers  (for  ex- 
ample, the  probability  of  a  male  birth  corresponds  to  the 
percentage  that  the  male  births  bear  to  the  total  number 
of  births). 

As  has  been  stated,  functions  of  probabilities  must  also 
be  considered.  An  example  of  such  is  the  ratio  of  male 
to  female  births,  that  is,  the  coordinate  number  indicating 
how  many  male  births  occur  to  1,000  female  births.  This  co- 
ordinate number  is  in  itself  not  a  probability  but  it  is  a  func- 
tion of  such,  that  is,  a  function  of  the  (analytic)  probability 
of  a  male  birth  in  relation  to  the  total  number  of  births. 
If  we  let  V  =  the  probability  of  a  male  birth,  p  =  the  ratio 

^*  Abhandlungen  zur  Theorie  der  Bevolkerungs-  und  Moralstatistik, 
IV,  "  tibersiclit  der  demographischen  Elemente  und  ihrer  Beziehungen 
zu  Einander,"  p.  62.  Cf.  also  von  Bortkiewicz,  Das  Gesetz  der  kleinen 
Zahlen,  p.  26,  and  Czuber,  Wahrscheinliclikeitsrechnung,  p.  302  f. 

"  Such  a  probability  applies  directly  to  the  success  or  failure  of 
an  event. 

"  Cf.  Abhandlungen  zur  Theorie,  etc.,  V,  "Uber  die  Ursachen  der 
geringen  Veranderlichkeit  statistischer  Verhaltniszahlen,"  p.  84. 


CLASSIFICATION  OF  STATISTICAL  SERIES        21 

of  the  male  to  the  female  births,  z  =  1,000,  p  =  the  number 
of  boys  bom  to  1,000  girls,  then  the  equation  z  =  ^-^~ 
follows.  If  on  the  average  515  males  occur  in  every  1,000 
births,  then  the  probability  of  a  male  birth  (v)  =  0.515 
and  z  =  1,062,  that  is,  1,062  males  are  born  for  every  1,000 
females. 

Kelative  numbers,  which  do  not  take  the  form  of  numeri- 
cal probabilities  and  are  not  functions  of  such,  are  desig- 
nated by  Lexis  as  "  Koordinations-verhaltnisse  "  (coordi- 
nate relations).  "  They  are,  in  general,  relations  between 
statistical  totals,  which  are  independent  from  each  other 
either  in  whole  or  in  part.  To  this  class  belong  the  various 
coefficients  (death  or  marriage  coefficients),  and  relations 
such  as  the  yearly  number  of  births  to  marriages,  etc. ' '  " 

The  peculiar  feature  of  relative  numbers,  which  appear 
as  numerical  probabilities  or  functions  of  such,  consists  in 
the  fact  that  the  methods  of  the  theory  of  probabilities  may 
be  applied  to  them,  while  such  methods  may  not  be  applied 
to  other  relative  numbers.  The  theory  of  probability  offers 
first  of  all  a  means  of  determining  the  reliability  of  numer- 
ical probabilities,  that  is,  of  establishing  between  what  limits 
the  theoretical  probability,  which  gives  rise  to  the  observed 
values,  probably  lies.  Since  the  reliability  of  an  observed 
probability  varies  directly  with  the  number  of  observations 
upon  which  it  is  based,  probabilities  based  upon  larger  sta- 
tistical masses  have  more  scientific  value  than  those  based 
upon  smaller  ones.  Numerical  probability  determined  for 
a  totality  is,  however,  the  arithmetic  mean  (simple  or 
weighted)  of  corresponding  values  for  its  portions.  Ac- 
cordingly, the  theory  of  probability  under  definite  condi- 
tions affords  a  particular  raison  d'etre  for  the  arithmetic 
mean  of  certain  items.  Furthermore,  it  is  possible  by  means 
of  the  theory  of  probability  to  compare  several  numerical 
probabilities  as  to  whether  the  difference  between  them  may 
probably  be  ascribed  to  chance  or  whether  we  must  assume 

»Ubid.   p.  84  f, 


22  STATISTICAL  AVERAGES  IN  GENERAL 

that  unequal  theoretical  probabilities  are  involved  in  the 
two  observations;  this  latter  ease  would  indicate  essential 
differences  in  the  causes  of  the  phenomena.  Finally,  the 
theory  of  probability  gives  a  criterion  for  measuring  the 
dispersion  of  a  series  of  numerical  probabilities,  especially 
for  determining  whether  the  various  members  of  the  series 
may  in  fact  be  regarded  as  empirical  determinations  of 
the  same  theoretical  probability,  or,  as  the  case  may  be,  of 
a  theoretical  probability  affected  only  by  accidental  changes. 

More  recent  authors  like  Lexis  and  Bortkiewicz,  it  is 
true,  go  so  far  as  to  assert  that  the  various  relative  numbers,  • 
even  if  they  satisfy  the  formal  conditions  in  question,  may 
not  be  summarily  treated  as  approximate  values  of  proba- 
bilities. They  claim  that  the  question,  whether  definite 
relative  numbers  may  be  regarded  as  empirical  numerical 
probabilities,  can  only  be  answered  in  the  affirmative,  if 
these  relative  numbers  belong  to  a  series  of  values  whose 
dispersion  follows  the  theory  of  probability. 

Very  similar  to  the  series  of  relative  numbers  are  the 
series  whose  members  are  averages  for  an  individual  meas- 
urement or  other  quantitative  observation.  Such  series  are 
not  frequent,  but  do  occasionally  occur.  A  large  mass  of 
quantitative  observations  may  be  subdivided  following  a 
criterion  of  space,  time,  quantity  or  quality,  and  accord- 
ingly particular  averages  may  be  computed  for  the  parts 
and  represented  in  a  series.  Thus,  for  example,  the  age 
data  at  time  of  marriage  or  death  may  be  subdivided  and 
particular  values  computed  for  the  average  age  of  those 
marrying  or  dying  in  different  months  or  provinces  or 
occupations.  The  question  now  arises,  how  from  such 
series,  whose  items  are  themselves  averages,  we  may  com- 
pute the  general  average  which  is  necessary  for  different 
purposes,  especially  as  a  basis  of  comparison. 

The  parts,  to  which  the  averages  of  the  series  refer,  are 
not,  as  a  rule,  of  equal  size.  Thus  different  provinces  and 
different  occupations  generally  show  varying  numbers  of 


CLASSIFICATION  OF  STATISTICAL  SERIES        23 

deaths  and  marriages,  besides,  these  numbers  vary  from 
month  to  month.  Therefore  various  significance  or  weight 
attaches  to  the  items  of  the  series.  But  averages  for  an 
individual  observation  have,  like  relative  numbers,  the  char- 
acteristic of  not  expressing  the  magnitude  of  the  masses  to 
which  they  refer.  From  the  figure  for  an  average  age  it  is 
not  possible  to  deduce  the  number  of  persons  whose  age 
was  considered  in  the  computation  of  the  mean.  Hence, 
from  a  series  of  averages  it  is  impossible,  as  a  rule,  to 
compute  the  general  average  directly;  on  the  contrary,  we 
must  compute  the  more  comprehensive  average  of  the 
higher  order  independently  on  the  basis  of  all  the  individual 
cases  entering  into  the  various  subdivisions  making  up  the 
series. 

The  relations  existing  between  the  means  for  the  totality 
and  for  the  subdivisions  depend  upon  the  kind  of  averages 
composing  the  series.  The  general  average  of  a  series  of 
arithmetic  means  represents  the  weighted  average  of  such 
items.  If  the  original  individual  data  are  not  available, 
the  average  of  the  higher  order  may  be  computed  directly 
from  the  items  of  the  series,  by  combining  them  with 
properly  chosen  weights.  It  may  be  necessary  to  estimate 
the  weights.  If  the  items  of  the  series  refer  to  subdivi- 
sions of  equal  weight,  the  weighted  average  coincides  with 
the  simple  arithmetic  mean  of  the  items  and,  therefore,  we 
may  dispense  wth  weights. 

Of  course  not  only  arithmetic  means  but  also  other  aver- 
ages such  as  medians  or  modes  may  occur  in  the  series. 
It  may  be  instructive  to  compare  the  items  of  such  series 
with  a  general  average  (median,  mode,  etc.)  computed 
from  the  original  individual  data.  Thus  we  may  compare 
the  medial  ("  probable  ")  or  modal  ("  normal  ")  length 
of  life  of  different  sections  of  the  population  with  a  similar 
average  computed  for  the  total  population.  However,  there 
is  no  general  rule  for  the  relation  existing  between  the 
median  or  mode  of  a  whole  and  the  medians  or  modes, 


24  STATISTICAL  AVERAGES  IN  GENERAL 

respectively,  of  the  parts.^^  This  relation  will,  in  every 
case,  depend  on  the  manner  in  which  the  whole  has  been 
subdivided. 

*'  The  essentials  of  the  classification  of  statistical  series,  given 
above,  rests  upon  that  given  by  Edgeworth  and  Czuber.  They  have 
not  worked  out  the  details  but  their  work  suggests  them.  Edge- 
worth  says:  "Two  cases  may  be  distinguished:  1,  where  the  returns 
with  which  we  have  to  deal  are  measurements  in  space  or  time, 
e.  g.,  statures  of  men  or  ages  at  death,  and  mere  numbers,  e.  g., 
yearly  deaths;  or  2,  ratios;  such  as  that  of  male  to  female  births, 
or  rates  of  mortality"  ("On  Methods  of  Statistics,"  Jubilee  Volume 
of  the  Roy.  Stat.  Soc,  1885,  p.  188).  Czuber  says:  "As  aids  in 
the  description  of  human  phenomena  there  are,  besides  the  relative 
numbers  which  we  previously  considered  and  which  we  designated  as 
intensive  magnitudes  because  they  express  the  intensity  of  an  action 
or  the  frequency  of  a  phenomenon,  also  extensive  magnitudes  which 
are  expressed  in  familiar  units  (dollars,  yards,  years).  In  par- 
ticular, they  express  the  space  of  time  during  which  a  condition 
exists  or  they  measure  the  interval  between  two  changes  of  condi- 
tion and  are  partly  of  biological,  partly  of  sociological  and  economic 
interest.  As  illustrations  we  may  cite:  the  length  of  life  of  indi- 
viduals of  a  deceased  population;  the  present  length  of  life  of 
individuals  of  a  living  population;  the  ages  of  brides  and  grooms 
at  marriage;  the  duration  of  married  life;  the  duration  of  the 
activity  of  individuals  in  certain  occupations;  the  duration  of  sick- 
ness in  various  age  groups,  etc."  (Die  Wahrscheinlichkeitsrechnung 
und  ihrer  Anwendung  auf  Fehlerausgleichung,  Statistik  und  Lebens- 
versicherung,  1903,  p.  333). 


CHAPTER  II 

ISOLATED  AVERAGES  AND  STATISTICAL  EVOLUTION 
FROM  ISOLATED  AVERAGES  TO  AVERAGES  BASED 
UPON  SERIES  OP  ITEMS 

An  average  serves  to  characterize  a  number  of  divergent 
quantities  by  a  single  value.  These  quantities  usually  form 
a  series.  There  are,  however,  also  statistical  values  of  the 
character  of  averages,  which  do  not  originate  from  series, 
since  the  quantities,  which  the  average  properly  represents, 
are  not  known.  These  ''  isolated  "  averages  are  computed 
or  estimated. 

A.  ISOLATED  AVERAGES  OBTAINED  BY  COMPUTATION 

The  arithmetic  average  of  a  series  is  the  ratio  of  the 
sum  of  the  items  to  their  number.  In  ease  we  do  not 
know  the  single  items,  but  merely  their  sum  and  number, 
the  ratio  gives  an  *'  isolated  "  average.  Thus,  for  instance, 
the  average  wage  of  workmen  in  an  industry  may  be  com- 
puted, even  if  the  individual  wages  are  not  known,  in  ease 
the  total  wages  and  the  number  of  workmen  are  known. 
Of  course  a  complete  series  of  wage  data  would  give  vastly 
more  information  concerning  the  wage  status  of  the  work- 
men than  would  an  isolated  average.  The  series  would 
show  the  wage  conditions  in  detail;  wage  classes  could  be 
formed  and,  besides  the  arithmetic  mean,  other  means 
(median,  mode,  etc.)  might  be  determined.  Also  the  wage 
data  might  be  tabulated  according  to  sex,  age,  occupation, 
ad  lib.  Isolated  averages  for  magnitudes  which  permit 
of  individual  measurement  belong,  therefore,  to  a  rather 
primitive  stage  of  statistical  method.     Especially  in  the 

25 


26  STATISTICAL  AVERAGES  IN  GENERAL 

field  of  wage  statistics  the  development  has  generally  been 
from  the  computation  of  isolated  averages  to  the  complete 
tabulation  of  the  wages  and  computation  of  averages  from 
the  individual  observations.^^""* 

^'  For  instance,  an  isolated  average  wage  was  computed  by  divid- 
ing the  total  wages  by  the  number  of  workmen  in  Czornig's  "  Indus- 
triestatistik  der  osterreichischen  Monarchic  fiir  das  Jahr  1856." 
Likewise,  the  average  income  per  year  is  obtained  in  the  current 
Austrian  mine-workers'  wage  statistics  (published  annually)  by  divid- 
ing the  total  wages  paid  by  the  average  number  of  workmen.  This 
average  yearly  amount  received,  however,  is  differentiated  according 
to  the  district  and  occupation  of  the  workmen. 

Average  wages  are  also  obtained  in  the  United  States  census  by 
the  summary  methods  under  discussion.  The  total  amount  of  wages 
paid  and  the  average  number  of  workmen  are  asked  for  and  the 
average  wage  then  computed.  The  inadequacy  of  this  treatment  of 
wages  has  often  been  remarked  by  the  Census  Office.  The  objections 
to  the  method  are  given  at  length  in  Part  I  of  the  Report  on 
Manufactures  of  the  Twelfth  Census  (Vol.  VII,  pp.  cxi  and  cxii). 
Consequently,  a  special  wage  investigation  was  undertaken  in  con- 
nection with  the  census  of  1900,  in  which  detailed  wage  data  were 
obtained  for  a  large  number  of  industries  (see  Twelfth  Census, 
Special  Reports,  Employees  and  Wages,  1903). 

Numerous  statisticians,  Boehmert  in  particular,  have  designated 
the  purpose  of  wage  statistics  to  be  expressly  that  of  dealing  ade- 
quately with  individual  earnings.  Likewise,  the  International 
Statistical  Institute  expressed  a  similar  opinion  in  a  resolution 
adopted  in  1891  at  its  Vienna  session.  But  the  treatment  of  rates 
of  wages  (taux  de  salaires,  Lohnsatze)  also  possesses  great  economic 
significance.  The  number  of  workmen  receiving  certain  rates  of 
wages  may  be  ascertained  and  the  average  wage  rates  for  greater 
groups  of  workers  may  be  computed.  From  the  wage  rates  and 
length  of  time  worked  the  earnings  may  be  derived.  The  English 
and  American  statistics  are  frequently  based  upon  rates  rather  than 
upon  earnings.  The  above-mentioned  Report  on  Employees  and 
Wages  was  chiefly  based  upon  rates  of  wages.  Similarly,  the  wage 
statistics  of  the  Berlin  Statistical  Bureau,  published  in  1904,  had  to 
do  with  wage  rates  for  the  various  kinds  of  work  in  various  industries. 

^'a  Wage  groups  were  given  in  the  reports  of  the  census  of  1890. 
The  reports  of  the  bureaus  of  laJjor  of  Massachusetts,  New  Jersey, 
and  Kansas  also  present  wage  groups.    Bulletin  93  of  the  Bureau  of 


ISOLATED  AVERAGES  27 

In  other  cases  a  similar  development  is  impossible  and 
in  these  the  isolated  average  is  still  computed.  The  reason 
for  such  lack  of  development  is  either  that  it  is  impossible 
to  obtain  the  individual  measurements  or  that  to  do  so 
would  be  inquisitorial  on  the  part  of  the  state.  Thus  we 
generally  compute  per  capita  consumption  of  meat,  beer, 
tobacco,  etc.,  as  isolated  averages.^"  However,  if  we  could 
obtain  the  complete  series  representing  the  individual  con- 
sumption of  meat,  beer,  tobacco,  etc.,  new  and  important 
facts  of  value  to  the  economist  and  hygienist  would  un- 
doubtedly be  ascertained.  All  those  individuals  who  do 
not  consume  these  articles  and  who  strongly  influence  the 
general  average  could  be  excluded;  the  average  per  capita 
consumption  could  be  found  for  the  remainder  of  the  popu- 
lation, which  could  also  be  subdivided  according  to  the 
degrees  of  consumption.  It  is  not  profitable  to  press  these 
points,  however,  as  these  statistics  cannot  be  ascertained.^^ 
In  consumption  statistics  we  must,  therefore,  be  satisfied 
with  isolated  averages.  Similar  averages,  likewise,  are  to 
be  found  in  other  fields.  Thus  we  compute  the  number  of 
letters,  periodicals,  or  money  orders  per  head,  or  lottery 
stakes  per  head.  Although  the  individual  statistics  might 
be  of  interest  it  is  impossible  to  obtain  them. 

Besides  the  averages  which  represent  a  series  of  measure- 

the  Census  gives  the  earnings  of  wage-earners  in  1905  in  groups. 
The  last  investigation  referred  to  is  probably  the  most  reliable  ex- 
tensive investigation  of  wages  ever  undertaken  in  the  United  States. 
The  wages  of  over  three  million  wage-earners  in  manufacturing  in- 
dustries are  tabulated  in  a  frequency  table.  Various  state  bureaus, 
and  the  Interstate  Commerce  Commission  as  well,  give  averages  and 
nothing  else.  The  usage  in  the  United  States  as  to  the  collection 
of  rates  of  wages  or  earnings  is  not  at  all  uniform. — Translator. 

'°  This  computation  is  made  by  dividing  the  total  amount  con- 
sumed in  a  country  (production  plus  imports  minus  exports)  by 
the  population. 

*^  A  start  has  been  made  toward  statistics  of  individual  consump- 
tion by  the  collection  and  statistical  treatment  of  family  budgets 
and  account  books,  upon  which  modern  statistics  lays  great  stress. 


28  STATISTICAL  AVERAGES  IN  GENERAL 

ments,  possible  but  not  actually  found,  because  of  obstacles 
of  a  statistical  character,  there  are  numerous  averages  for 
magnitudes  which  are  incapable  of  individual  observation 
of  any  sort.  Of  the  latter  type  of  averages  are  average 
air  space  or  floor  space  per  inhabitant  of  a  tenement,  the 
average  debt  of  a  country  per  capita,  the  per  capita  cost 
of  national  and  city  administration,  the  per  capita  expendi- 
tures or  receipts  of  the  government,  the  average  value  or 
quantity  of  foreign  trade  per  capita,  etc. 

In  all  of  these  cases  individual  measurements  are  in- 
conceivable. It  is  impossible  to  measure  a  definite  air  space, 
portion  of  the  government  debt,  or  cost  of  administration 
for  each  individual.  In  such  cases  we  are  not  dealing  with 
an  "  isolated  average  "  for  a  "  potential  element  of  meas- 
urement," but  with  a  relative  number  which  originates 
through  interrelating  two  wholly  independent  magnitudes. 
We  should  not  speak  of  such  a  ratio  as  an  average,  since  in- 
dividual measurements  are  impossible,  but  rather  as  the 
size-ratio  of  two  magnitudes. 

Such  size-ratios  are  very  common  in  all  branches  of 
statistics.  The  most  varied  masses  may  be  interrelated  for 
special  purposes.  In  each  case  the  number  of  units  in 
one  mass  is  divided  by  the  numbers  of  units  in  the  other. 
Thus  the  division  of  the  deaths  during  a  year  by  the  mean 
population  gives  the  death  coefficient.  If,  as  is  frequently 
done,  the  quotient  is  then  multiplied  by  100  or  1,000,  the 
relative  number  indicates  how  many  units  of  the  first 
mass  there  are,  on  an  average,  to  100  or  1,000  units  of 
the  second  mass.  For  example,  the  death  rate  (that  is, 
the  death  coefficient  multiplied  by  1,000)  indicates  how 
many  deaths  occur  on  an  average  to  1,000  living  of  the 
mean  population.^^     In  a  similar  way,  it  may  be  computed 

"'  While  mortality  coefficients  as  well  as  death  rates  originate 
through  interrelating  the  number  of  deaths  and  the  mean  popula- 
tion, differing  from  each  other  only  in  the  position  of  the  decimal 
point,  the  probability  of  death  is  computed  by  dividing  the  number 


ISOLATED  AVERAGES  29 

how  many  births,  marriages,  crimes,  suicides,  etc.,  occur 
on  an  average  to  every  1,000  or  10,000  living  of  the  mean 
population.  Single  values  for  concrete  groups  of  1,000  or 
10,000  persons  each  cannot  be  obtained,  since  the  mean 
population  is  an  abstraction  which  cannot  be  submitted, 
either  wholly  or  in  part,  to  a  constant  direct  observation. 

Likewise  the  population  (or  definite  groups  of  the  pop- 
ulation) and  the  area  are  frequently  interrelated.  The  re- 
sult gives  the  average  population  per  square  mile  (density 
of  population).  To  determine  the  population  of  the  in- 
dividual square  miles  would,  of  course,  be  impossible,  since 
these  areas  are  units  of  computation  and  not  objective 
values.  Similarly  the  average  number  of  schools,  post- 
offices,  etc.,  the  average  length  of  road,  railroad,  telegraph 
line,  etc.,  is  computed  for  some  round  number  of  square 
miles.  We  have  also  to  do  with  size-ratios  when  railroad 
statistics  give  the  freight  density  (ton-miles  per  mile  of 
line)  or  when  labor  statistics  give  the  average  number  of 
applicants  for  each  place  offered. 

Not  only  statistical  coordinate  numbers,  ,out  also  the 
equally  common  subordinate  numbers,  appear  regularly  in 
the  form  of  averages.  We  say:  in  100  infants  there  are  on 
an  average  51  boys  and  49  girls;  of  1,000  inhabitants,  ac- 
cording to  the  census,  so  many  belong,  on  an  average,  to  the 
different  nationalities,  religions,  occupations,  etc. ;  of  1,000 
individuals  of  the  same  age,  so  many  die,  on  an  average, 
at  specified  ages  (mortality  table).  Such  subordinate  num- 
bers express  the  composition  of  larger  statistical  masses 
in  simplified  form  by  reducing  the  total  to  100  or  1,000. 
Concrete  groups  of  100  or  1,000  units  each,  of  course,  do 
not  exist. 

It  is  evident,  therefore,  that  those  statistical  quotients, 
which  refer  to  virtual  elements  of  measurement  are,  indeed, 

of  deaths  by  the  total  popuhition  among  which  the  deaths  occur 
during  the  period  of  years  in  question  and  thus  the  latter  differs 
from  the  former  both  in  conception  and  numerical  value. 


30  STATISTICAL  AVERAGES  IN  GENERAL 

true  even  though  they  are  ''  isolated  "  averages,  but  that 
the  much  more  numerous  coordinate  and  subordinate  num- 
bers, although  they  appear  as  "  averages  "  in  statistical 
language,  are  not  actually  averages  in  the  literal  sense  of 
the  word.  Thus  the  average  number  of  inhabitants  per 
square  mile  is  not  an  average  of  the  population  of  each 
square  mile  of  the  country,  since  the  determination  of  such 
numbers  is  inconceivable. 

But  among  statistical  relative  numbers  there  are  some 
which  are  not  merely  nominal  averages  but  are  real  aver- 
ages, though,  to  be  sure,  in  a  new  sense.  We  shall  now 
consider  these  especially  interesting  relative  numbers. 

In  the  preceding  section  ^^  we  found  that  the  arithmetic 
average  for  certain  time,  space,  qualitative,  or  quantitative 
series  must  be  computed  as  a  relative  number  of  a  higher 
order.  Such  relative  numbers  are,  therefore,  true  averages 
wdth  respect  to  the  items  referring  to  parts  of  the  totality. 
Relative  numbers  are  in  their  nature  averages  even  though 
there  be  no  special  relative  numbers  referring  to  parts  of 
the  totality,  on  the  condition,  that  such  special  relative 
numbers  are  theoretically  possible.  This  condition  is  fre- 
quently fulfilled. 

Two  examples  will  make  this  clear.  Statisticians  know 
that  the  sex-ratio  depends  upon  the  density  of  population 
of  the  district  (in  consequence  of  selective  migration),  and 
upon  the  age  class  (in  consequence  of  the  varying  mortality 
of  the  sexes).  The  general  sex-ratio  for  the  total  popula- 
tion is,  therefore,  a  mean  originating  from  divergent  sex- 
ratios  for  different  places  and  age  classes,  possessing  in 
itself  the  character  of  an  average,  even  if  the  sex-ratios 
for  the  various  subdivisions  are  not  known.  Similarly  the 
annual  death  rate  is  an  average,  since  the  mortality  may 
fluctuate  during  the  year.  This  average  levels,  therefore, 
the  "  time  frequency  "  of  mortality.^*    It  levels,  also,  the 

^»  Cf.  p.  18  f. 

^*  Compare   Mischler's   Handbuch   der   Verwaltungsstatistik,    §  30, 


ISOLATED  AVERAGES  31 

mortality  differences  which  usually  exist  for  different  geo- 
graphic divisions,  and  for  different  occupations,  age  classes, 
and  the  like. 

Suppose  we  assume  that  mortality  varies  with  sex,  age, 
and  civil  condition.  A  general  death  rate  is  the  average 
of  "  special  "  death  rates  for  various  groups  divided  ac- 
cording to  sex,  age,  or  civil  condition.  Likewise,  the  ' '  spe- 
cial" death  rates  for  the  groups  are  themselves  averages, 
as  there  are  numerous  other  influences  affecting  mortality. 
For  example,  the  death  rate  of  individuals  of  a  certain 
sex,  age,  and  civil  condition  is  an  average  of  the  various 
death  rates  of  the  subdivisions  of  such  individuals  accord- 
ing to  occupation.  But  there  are  many  other  things  in- 
fluencing mortality,  such  as  economic  condition,  nourish- 
ment, housing,  the  altitude,  geological  formation,  etc.  Since 
a  death  rate  referring  to  an  entirely  homogeneous  group 
of  the  population  does  not  exist,  every  death  rate  must  al- 
ways be  thought  of  as  an  average  leveling  the  divergent 
rates  of  component  parts  of  the  population  in  question. 

What  holds  for  death  rates  holds  as  well  for  most  of 
the  remaining  demographic  relative  numbers,  marriage 
rates,  birth  rates,  and  the  life.  Demographic  phenomena 
show,  as  a  rule,  time  and  space  fluctuations  which  vary 
in  degree  among  different  groups  of  the  population.  Age, 
sex,  civil  condition,  occupation,  and  economic  condition 
always  have  their  influence.  Some  phenomena,  such  as  sui- 
cide, show  the  influence  of  religion;  life  in  cities  produces 
other  effects  than  life  in  the  country.  Quetelet  sought  in 
his  Social  Physics  to  ascertain  the  action  of  the  factors 
influencing  demographic  phenomena  and  numerous  other 
writers  of  a  later  date  have  presented  "  Schemes  of  In- 
fluences Affecting  Mankind,"  or  statistical  "  Sj'stems  of 
Causes."" 

"  Die  Massenerscheingungen  als  Funktion  der  Zeit,"  especially, 
p.  90  f. 

'°  See,  for  example,  in  this  connection:   Ottingen's  Moralstatistik, 


32  STATISTICAL  AVERAGES  IN  GENERAL 

The  fact  not  always  sufficiently  regarded,  that  relative 
numbers  are  themselves  frequently  averages,  is  of  the 
greatest  importance  for  statistics.  Relative  numbers  should 
above  all  make  comparisons  possible.  For  this  purpose 
we  compare  the  death  rates  of  diiferent  years  or  classes 
of  the  population.  But  there  is,  as  will  be  subsequently 
shown,  generally  an  element  of  uncertainty  in  the  com- 
parison of  averages  and  hence  of  relative  numbers.  Thus 
the  difference  in  the  general  death  rates  of  two  countries 
may  be  caused  simply  by  the  different  composition  of  the 
population  of  the  two  countries,  while  the  death  rate  in  the 
individual  population  groups  or  age  classes  in  each  country 
may  be  the  same.  A  country  with  a  larger  number  of  chil- 
dren will  by  reason  of  this  very  fact  show  a  larger  general 
death  rate  than  a  country  with  a  small  number  of  children, 
even  though  the  death  rate  in  the  corresponding  age  classes 
of  the  two  countries  is  the  same. 

Engel's  Bewegung  der  Bevolkerung  im  Konigreiche  Saehsen  in  den 
Jahren  1834-1850,  Wagner's  Gesetzmassigkeit  in  den  scheinbar 
willkiirlichen  menschlichen  Handlungen,  Gabaglio's  Storia  e  Teorica 
generals  della  Statistica,  and,  most  recently,  Colajanni's  Statistica 
teorica. 

Quetelet  differentiated  between  "  natural  "  and  "  accidental  "  or 
"  perturbing "  influences,  the  latter  originating  from  man  himself. 
As  the  "  natural "  influences  affecting  mortality  he  named :  the 
influence  of  locality,  sex,  age,  the  type  of  year,  the  seasons,  the 
time  of  day,  and  the  influence  of  various  diseases ;  "  accidental  "  or 
"  perturbing "  causes  of  death  are :  the  influence  of  occupation,  eco- 
nomic condition,  morality,  enlightenment,  and  the  political  and  re- 
ligious development.  The  influences  enumerated  by  Quetelet  are 
more  or  less  generally  recognized;  their  division  into  the  two  groups 
"  natural  "  and  "  accidental  "  or  "  perturbing  "  has,  however,  been 
proven  invalid. 

Colajanni  enumerates,  first,  physical  influences,  such  as  climate, 
soil,  moisture,  etc.;  second,  anthropological  causes,  such  as  sex,  age, 
mental  and  physical  constitution,  inherited  traits,  and  race;  third, 
social  causes,  such  as  density  of  population,  relations  brought  about 
by  living  in  groups,  divisions  of  the  country,  laws,  political  organ- 
ization, stage  of  culture,  religion,  family  position,  and  occupation. 


ISOLATED  AVERAGES  33 

There  are,  accordingly,  often  differences  of  opinion  about 
the  conclusions  to  be  drawn  from  a  comparison  of  relative 
numbers.  Special  methods  have  been  invented  for  making 
a  comparison  of  definite  relative  numbers  possible. ^^  The 
most  suitable  for  purposes  of  comparison  are,  as  will  be 
shown  in  detail  later,  those  relative  numbers  which  refer 
to  elements  containing  the  least  possible  differences.  These 
are  relative  numbers  which  are  computed  for  most  nearly 
homogeneous  masses.  Such  relative  numbers  possess,  com- 
paratively, the  greatest  scientific  value,  and  are  therefore 
one  of  the  chief  aims  of  modern  statistics.  For  this  reason, 
those  relative  numbers  which  are  to  be  regarded  as  aver- 
ages are  often  resolved  into  more  homogeneous  compo- 
nents, which  may  then  be  compared,  so  to  speak,  as  special 
values  with  the  original  relative  numbers.  Thus  special 
death  rates  for  different  age  classes,  occupations,  etc.,  are 
being  computed  and  then  compared  as  special  values  with 
the  general  death  rate  for  the  whole  population.  It  is 
then  possible  in  such  a  case  to  compare  the  original  relative 
number  with  the  supplementary  special  values  and  in  this 
way  to  test  its  scientific  value  and  its  applicability  for 
different  statistical  methods. 

B.     ESTIMATED  ISOLATED  AVERAGES 

Isolated  averages  may  be  estimated  as  well  as  computed. 
In  any  case  the  individual  items  are  lacking  and  the  com- 
parison of  the  average  with  them  is,  therefore,  impossible. 
But  whereas  the  computed  average  is  fixed  numerically  the 
estimated  average  is  determined  between  limits  of  greater 
or  less  distance  apart.  Still,  as  a  matter  of  fact,  the  statis- 
tician is  not  infrequently  forced  to  estimate  averages  be- 
cause of  lack  of  the  necessary  data  for  computation.     The 

="  Thus,  for  example,  the  method  of  the  standard  population  serves 
the  purpose  of  making  the  death  rates  of  various  countries  com- 
parable  (see  p.  159  f.). 


34  STATISTICAL  AVERAGES  IN  GENERAL 

estimation  of  averages  for  "  potential  individual  observa- 
tions," and  the  estimation  of  relative  numbers,  which  are 
really  averages,  are  of  especial  importance, 

(a)  Estimation  of  the  Average  Size  of  a  Virtual  Individual 
Element  of  Observation 

In  a  large  number  of  cases  where  it  is  impossible  to 
obtain  the  individual  items  in  detail  and  where  an  isolated 
arithmetic  mean  cannot  be  computed,  the  problem  of 
estimating  that  mean,  or  other  mean  such  as  the  mode,  is 
intrusted  to  experts.  This  procedure  is  the  common  one 
in  ascertaining  the  average  wage  of  agricultural  laborers.^^ 
In  Austria  and  Germany  the  estimation  of  the  customary 
wage  is  necessary  for  the  administration  of  workingmen's 
insurance.  By  "  customary  wage  "  is  meant  the  wage 
received  by  the  greatest  number  of  laborers,  that  is,  the 
"  normal,"  prevailing,  predominant,  or  modal  wage.^® 

'^  See,  for  example,  the  collections  of  agricultural  wages  in  Aus- 
tria made  in  1895  and  1897  by  the  Landes  Kulturrate  and  Land- 
wirtschaftgesellschaften  (Ost.  Stat.,  Vol.  XLIV,  No.  1,  and  Stat. 
Mdhatsschrift,  1904,  p.  466  f. ).  In  many  cases  the  experts  have 
given  the  maximum  and  minimum  rates  as  well  as  the  arithmetic 
mean. 

"  Section  6  of  the  Austrian  laws  of  March  30,  1888,  R.  G.  Bl. 
No.  33,  concerning  sick  insurance  of  workmen,  contains  the  pro- 
vision that  the  aid  given  during  sickness  should  be,  at  least,  "60% 
of  the  daily  wages  customary  in  the  jurisdiction  received  by  laborers 
entitled  to  insurance  benefits."  Section  7  provides :  "  The  amount 
of  the  daily  wages  customary  in  each  "  isdiction  and  received  by 
laborers  entitled  to  insurance  benefits  are  to  be  fixed  periodically 
by  the  political  authorities  after  a  hearing  of  proxies  and,  in  those 
places  where  the  jurisdictions  have  representatives,  after  consultation 
with  committees  from  the  respective  jurisdictions.  If  it  is  found  that 
wages  vary  greatly  then  the  customary  daily  wages  of  several 
categories  may  be  established.  They  may  be  established  for  men, 
women,  children  and  youths,  especially,   ..." 

In  Germany  the  customary  wages  received  by  day  laborers  are 
established  for  the  various  communities  by  the  higher  administrative 
authorities  after  giving  the  local  authorities  a  hearing. 


ISOLATED  AVERAGES  35 

The  estimation  of  averages  is  also  of  importance  in  the 
railroad  business.  Since  railways  transport  much  freight 
without  weighing  each  item  it  is  necessary  to  fix  rates  upon 
the  basis  of  estimated  average  weights.  For  example,  the 
Austrian  railways  fix  the  following  average  weights:  a 
sucking  pig,  20  kg.;  a  young  boar,  30  kg.;  a  lean  hog,  60 
kg. ;  a  fat  hog,  170  kg.,  etc.-** 

Many  times  an  estimate  of  an  arithmetic  average  is  made 
in  order  that  such  estimated  average  may  be  multiplied 
by  the  number  of  elements  to  which  it  refers,  thus  giving 
the  sum  total  of  the  unknown  individual  elements.  Thus 
we  estimate  the  arithmetic  average  amount  of  money  taken 
out  of  the  country  by  an  emigrant  in  order  to  ascertain 
the  total  amount  taken  out,  which  is  the  product  of  the 
average  and  the  number  of  emigrants.^"  Similarly,  vari- 
ous writers  have  used  the  following  equation  in  estimating 
national  income: 

(Total  of  incomes  reported  subject  to  tax)  +  (Estimated 
average  of  untaxed  incomes)  (Number)  =  National  Income. 

Estimates  of  the  population  of  the  past  depend  upon 
preliminary  estimates  of  averages.  If  we  know  the  number 
of  families,  the  number  of  dwellings,  or  the  number  of 
hearths  in  a  city  or  country  and  can  estimate  the  average 
number  of  persons  per  family,   dwelling,  or  hearth,  the 

*°  These  arithmetic  average  weights,  which  ought  to  correspond  to 
the  actual  weights,  must  be  disting^aished  from  the  "  normal  weights  " 
("  Xormalgewichte  ")  of  railroad  tariffs,  which  differ  greatly  from 
the  actual  weights  and  are  merely  assigned  to  serve  in  certain  rail- 
road tariff  computations. 

""  Cf.  the  Tabellen  zur  Wahrungsstatistik  issued  by  the  Aus- 
trian Minister  of  Finance,  2nd  ed.,  Pt.  II,  Vol.  Ill,  §  13,  "  Daten 
zur  Zahlungsbilanz,"  p.  823,  where,  for  example,  it  is  assumed  that 
the  Austrian  emigrants  to  Brazil,  Argentine  Republic,  and  Canada 
take  with  them,  on  an  average,  100  crowns  in  ready  money.  On 
p.  832  of  the  same  publication  it  is  estimated  in  a  similar  way — 
naturally  upon  the  data  from  certain  stopping  places — that  the 
arithmetic  average  daily  expenditures  of  foreigners  stopping  in  Aus- 
tria are  15  cro^vns. 


36  STATISTICAL  AVERAGES  IN  GENERAL 

product  of  corresponding  numbers  gives  the  population  of 
the  area  in  question.  Methods  similar  to  those  just  quoted 
must  be  used  in  computing  the  present  population  of  coun- 
tries where  a  census  is  not  taken.^^ 

The  statistics  of  the  value  of  imports  and  exports  de- 
pend, in  most  countries,  upon  estimates  of  experts.  Like- 
wise, estimates  of  average  weights  are  necessary.  In  order 
to  express  all  imported  and  exported  wares  as  a  total, 
they  must  be  expressed  in  the  same  unit  of  weight.  In 
the  case  of  those  articles  which  are  recorded,  not  by  weight, 
but  by  the  piece  or  otherwise,  it  is  necessary  to  estimate 
the  average  weight.  Thus,  in  the  statistics  of  the  foreign 
trade  of  Austria-Hungary,  cattle,  hats  (with  certain  ex- 
ceptions), carriages,  bicycles,  watches,  etc.,  are  not  reported 
according  to  weight  but  by  number.  Accordingly,  in  the 
year  1904  the  average  weight  of  an  ox  was  taken  as  450, 
650,  or  500  kg.,  depending  upon  whether  it  was  imported, 
exported,  or  shipped  in  domestic  trade. 

Many  statisticians  have  also,  especially  in  recent  times, 
tried  by  means  of  estimated  averages  to  compensate  for 
lack  of  statistics  of  production.  Thus  Czornig  computed 
the  total  production  of  Austria  from  the  ascertained  num- 
ber of  different  machines,  by  ascribing  to  them  a  definite 
average  productivity.    He  says  in  his  preface  to  the  In- 

»*  See  especially  the  report  submitted  by  Marcus  Rubin  at  the 
ninth  session  of  the  International  Statistical  Institute  (1903)  en- 
titled "  Sur  les  exploitations  dSmographiques  a  executer  dans  les  pays, 
oil  il  n'existe  pas  encore  de  recensements."  Prof.  L.  Gumplowicz 
relates  in  his  Verwaltungslehre  that  Lord  Macartney,  a  British 
ambassador,  computed  the  population  of  a  Chinese  province  from 
a  certain  store  of  salt  which  he  was  told  was  to  cover  the  con- 
sumption for  one  year.  Lord  Macartney  estimated  the  arithmetic 
average  consumption  per  capita  and  then  computed  the  number  of 
persons  that  could  be  supplied  with  the  given  store  of  salt  at  the 
average  consumption  estimated.  Westergaard  (Die  Grundziige  der 
Theorie  der  Statistik,  p.  270)  mentions  an  estimate  of  population 
upon  the  basis  of  grain  consumption  made  by  Crome  in  1785  (Grosse 
und  Bevolkerung  der  europiiischen  Staaten). 


ISOLATED  AVERAGES  37 

dustriestaUstik  der  osferreicJiischen  Monarchie  fiir  das  Jahr 
1856  (p.  vi)  :  "  In  each  ...  (of  the  different  branches  of 
industry)  .  .  .  there  is  a  teclinical  unit,  which  serves  as  the 
measure  of  production,  as,  for  instance,  the  loom  in  weav- 
ing, the  machine  or  the  vat  in  the  manufacture  of  paper,  the 
furnace  in  the  manufacture  of  steel,  the  number  of  work- 
men in  the  manufacture  of  machines,  etc.,  and  any  expert, 
knowing  this  technical  unit  of  an  industry,  will  be  able 
to  estimate  its  production  with  approximate  correctness." 
This  method  of  computing  the  production  by  multiplying 
the  number  of  machines  by  their  average  output,  which 
Block  also  has  recommended  in  his  Traite  de  Statistique 
(2nd  edition,  1886),  is,  however,  for  obvious  reasons  un- 
reliable, and  is  therefore  no  longer  used.^- 

Frequently,  estimates  of  averages  refer  to  magnitudes 
which  are  not  only  removed  from  observation  in  the  mass, 
but  which  even  individually  cannot  be  measured  exactly. 
Thus,  various  writers  have  tried  to  estimate  the  average 
capital  expended  on  the  education  of  an  adult  in  different 
classes  of  society  and  hence  to  express  in  figures  the  * '  value 
of  the  individual.  "^^ 

As  already  stated,  various  kinds  of  averages,  such  as  the 
arithmetic  mean,  the  median,  the  mode,  etc.,  may  be  com- 
puted from  series  of  individual  observations.  Hence,  not 
only  the  estimate  of  the  arithmetic  average  but  also  that 
of  the  relatively  most  frequent  magnitude  may  be  at- 
tempted. But  in  practice  the  various  kinds  of  averages 
are  not  always  sufficiently  distinguished;  those  whose  duty 
it  is  to  make  the  estimate  frequently  do  not  have  strict 

•^  Lavoisier  applied  a  still  more  questionable  method  in  1790  when 
he  utilized  the  number  of  plows,  found  in  some  way  or  other,  and 
the  estimated  average  performance  of  a  plow  to  derive  the  size  of 
the  fields  not  registered  and  the  amount  of  the  agricultural  pro- 
duction of  France. 

"See  Dr.  E.  Engel,  Der  Wert  des  Menschen  (1883),  Pt.  I,  "Der 
Kostenwert  des  Menschen"   (with  bibliography). 


38  STATISTICAL  AVERAGES  IN  GENERAL 

orders  and  choose  the  average  whose  determination  is  easiest 
for  them.  Now  it  is  a  special  characteristic  of  the  mode 
that  it  may  be  estimated  more  readily  than  the  arithmetic 
mean.  To  determine  the  latter,  all  the  single  cases  must 
be  taken  into  account  according  to  their  magnitudes;  but 
if  they  are  known,  the  average  is  computed,  not  estimated. 
The  mode  of  a  set  of  values,  on  the  other  hand,  easily  im- 
presses the  observer,  and  any  expert  will  be  able  to  give 
it  without  computation.  It  may  be  surmised,  therefore, 
that  estimated  averages  are  oftener  modes  than  arithmetic 
means. 

The  arithmetic  mean  and  the  mode  do  not,  as  a  rule, 
coincide.  This  fact  becomes  of  great  significance  as  soon 
as  the  sum  total  of  all  the  items  for  all  the  single  cases  is 
to  be  computed  by  means  of  the  estimated  average.  This 
sum  may  be  obtained  by  multiplying  the  number  of  cases 
by  the  average  value  of  the  items,  but  not  by  multiplying 
such  number  by  the  mode,  unless  this  latter  chances  to 
coincide  with  the  arithmetic  mean.  For  instance,  if  the 
average  size  of  a  family  is  multiplied  by  the  number  of 
families  the  population  is  obtained,  but  not  if  the  mode  is 
multiplied  by  the  number  of  families.  Accordingly,  esti- 
mated averages  for  a  potential  element  of  observation, 
when  used  to  compute  a  totality,  give  a  correct  result  only 
when  they  correspond  to  the  arithmetic  mean  and  not  to  the 
mode. 

Estimated  averages,  naturally,  do  not  possess  the  same 
value  as  averages  computed  from  detailed  data,  since  those 
who  make  the  estimate,  as  a  rule,  know  only  a  small  part 
of  the  individual  cases  and,  perhaps,  give  a  judgment  from 
limited  observations.  Therefore,  modern  statisticians  try, 
as  far  as  possible,  to  dispense  with  estimates  of  a  potential 
element  of  observation. 

It  is  to  be  observed  that  estimated  values  occur  some- 
times without  being  evident.  In  such  cases  there  is  dan- 
ger of  ascribing  to  the  value  in  question  a  greater  reliability 


ISOLATED  AVERAGES  39 

than  it  actually  possesses.  Thus,  for  example,  the  method 
of  direct  questioning  about  individual  average  wages,  often 
used  because  of  its  simplicity,  leads  generally  to  mere  esti- 
mates of  averages.  If  an  employer  is  asked  to  give  the  aver- 
age weekly  earnings  of  his  employees,  he  is  generally  able 
to  compute  them  correctly  from  a  series  of  successive  pay- 
days. But  it  is  unlikely  that  he  will  take  this  trouble.  He 
is  more  apt  to  attempt  a  more  or  less  arbitrary  estimate.^* 

(b)  Estimation  of  Relative  Numbers  which  are  Themselves 

Averages 

Relative  numbers  having  the  character  of  averages  are 
also  often  estimated  when  there  are  no  sufficient  data  for 
their  computation.  The  ratio  of  two  masses  is  often 
determined  for  the  purpose  of  computing  the  size  of  an 
unknown  mass  from  a  known  one.  Statisticians  often  em- 
ploy such  estimates  in  order  to  obtain  the  population  figures 
of  past  times.  If  we  know,  historically,  the  number  of 
citizens,  or  artisans,  or  slaves  of  a  city  or  country,  we  may 
estimate  what  percentage  of  the  entire  population  these 
various  classes  probably  formed  on  an  average  at  the  time 

'*  Individual  average  wages,  in  the  sense  explained  above,  were 
asked  for  at  the  inquiry  of  the  Reichenberg  Handels-  und  Gewerbe- 
kammer  in  1888  (see  Nordbohmische  Arbeiterstatistik,  Reichenberg, 
1891).  In  the  investigation  concerning  the  conditions  of  workmen 
in  the  Ostrau-Karwin  coal  district,  which  was  undertaken  by  the 
Austrian  Bureau  of  Labor  Statistics  in  1901  under  the  direction  of 
Dr.  Mataja,  the  actual  earnings  of  mine  workers  were  determined; 
arithmetic  average  wages  were  collected  solely  for  comparison  with 
the  arithmetic  average  wages  collected  from  workmen  of  the  dis- 
trict employed  in  other  occupations;  the  question  called  for  the 
amount  each  workman  received,  on  an  average,  per  week  during 
the  first  half  of  the  year  1901.  (See  Arbeiterverhaltnisse  im  Ostrau- 
Karwiner  Steinkohlenreviere.  Published  by  the  k.  k.  Arbeitsstatis- 
tisches  Amt  im  Handelsministerium.  Pt.  I,  "  Arbeitszeit,  Arbeits- 
leistungen,  Lohn-  und  Einkommensverhaltnisse,"  Vienna,  1904,  p. 
xlix. ) 


40  STATISTICAL  AVERAGES  IN  GENERAL 

in  question,  and  hence  we  may  compute  the  total  popula- 
tion. 

The  ratio  between  definite  statistical  masses  is  often 
fairly  constant  and  can  easily  be  estimated  to  be  within 
certain  limits.  This  holds  good,  for  instance,  for  the  rela- 
tion between  population  and  births  or  deaths.  Since  the 
time  of  Halley  and  John  Graunt,  innumerable  attempts 
have  been  made  to  compute  the  population,  lacking  a  cen- 
sus, by  means  of  estimated  death  or  birth  coefficients  based 
on  data  concerning  the  movement  of  population.^^  This 
method  has  often  been  used  to  reconstruct  the  population 
of  past  times.  In  Roumania  it  is  still  employed.^®  Also  in 
states  where  questions  about  religion  are  not  asked  in  the 
census,  the  attempt  has  often  been  made  to  compute  the 
numbers  allied  with  the  various  denominations  from  the 
known  number  of  communicants  and  the  estimated  percent- 
age that  this  latter  number  bears  to  the  former.  In  Amer- 
ica even  the  number  of  seats  in  the  churches  is  used  in 
making  this  computation.  This  estimated  ratio  of  the  com- 
municants or  seats  to  the  total  number  in  the  denomina- 
tion is  just  as  much  an  average,  since  it  varies  from  place 
to  place,  as  are  the  general  sex-ratio  of  the  population  and 
the  general  death  rate. 

Every  estimate  must  be  regarded  as  simply  an  approxi- 
mate value.  It  may  be  quite  accurate  in  some  cases,  but 
there  is  no  certainty.  Therefore  the  number  computed  by 
means  of  an  estimated  relative  number  must  also  be  re- 
garded as  merely  approximate.  The  fact  that  the  relative 
numbers  in  question  are  averages  causes  particular  difficul- 
ties. If,  for  instance,  the  population  of  a  country  is  to  be 
inferred  from  the  number  of  deaths,  the  one  who  makes  the 

"  Sonnenfels  in  his  Grundsiitze  der  Politik  (Vienna,  1819,  Vol. 
I,  p.  32)  cites  a  number  of  methods  of  estimating  population  which 
are  based  on  the  fact  that  "  political  computation  infers  the  number 
of  people  from  relations  which  are  determined  by  experience." 

»°  Compare  G.  v.  Mayr,  Bevolkerungsstatistik,  p.  15. 


ISOLATED  AVERAGES  41 

estimate  usually  applies  a  death  rate  which  has  been  ob- 
served to  be  true  of  a  certain  section  of  the  country  or 
class  of  the  population  but  which  may  not  hold  good  for 
the  entire  country.  Modern  statisticians  endeavor,  there- 
fore, to  secure  data  sufficient  to  enable  them  to  dispense 
with  estimates,  though,  to  be  sure,  their  attempts  have  hith- 
erto been  only  partially  successful. 

C.  SOME  INSTANCES  OF  THE  PROGRESS  OF  STA- 
TISTICS AWAY  FROM  ISOLATED  AVERAGES  AND 
TOWARD  AVERAGES  BASED  ON  SERIES  OF  ITEMS 
(average  length  of  life,  length  of  marriage,  length  of  a 
generation,  number  of  children  per  family) 

As  has  been  mentioned,  the  tendency  of  modern  statistics 
is  to  obtain  not  isolated  averages  but  statistical  series 
from  detailed  observations.  From  these  series  averages  may 
be  computed  whose  reliability  and  cogency  may  be  deter- 
mined in  individual  cases  by  comparing  them  with  the 
original  items  of  the  series.  In  the  following  discussion 
additional  remarkable  examples  will  be  given  of  the  prog- 
ress from  isolated  averages  toward  averages  from  statistical 
series.  These  examples  differ  in  character  from  the  cases 
already  mentioned  and  must  therefore  be  treated  separately. 
In  the  cases  which  we  now  take  up  the  chief  question  is, 
which  of  several  statistical  series  furnishes  the  most  valu- 
able average,  methodologically  speaking.  This  question 
may,  under  certain  circumstances,  be  of  great  significance, 
since  series  of  different  kinds  naturally  produce  averages 
of  different  character ;  also  the  value  of  the  average  depends, 
of  course,  on  the  individual  values  contained  in  the  series. 

An  interesting  example  of  the  evolution  in  question  is 
furnished  by  the  method  of  computing  the  average  length 
of  life.  In  former  days  it  was  generally  computed  as  an 
isolated  average,  by  dividing  the  population  by  the  number 
of  births  or  the  number  of  deaths.  But  soon  doubts  began 
to  arise  about  the  correctness  of  this  average.     Nowadays 


42  STATISTICAL  AVERAGES  IN  GENERAL 

it  is  a  statistical  commonplace  that  the  quotient  of  the  popu- 
lation and  number  of  births  (or  deaths)  would  give  the 
average  length  of  life  correctly  only  provided  that  the  pop- 
ulation is  stationary.  In  such  a  case  one  might  argue: 
each  birth  replaces  a  living  being,  in  one  year  the  xth 
part  of  the  population  is  replaced  by  new  births,  in  x  years, 
therefore,  the  total  population  is  so  replaced :  that  is  to  say, 
the  average  length  of  life  is  x  years.  So  one  might  also  say 
of  a  stationary  population :  if  the  yth  part  of  the  popula- 
tion is  lost  by  death  each  year,  the  whole  population  must 
be  renewed  in  y  years,  and  accordingly  the  average  length 
of  life  must  be  y  years.  In  a  stationary  population,  of 
course,  x  equals  y.  But,  as  a  matter  of  fact,  there  is  no 
such  thing  as  a  stationary  population ;  x  and  y  are  always 
different,  and  neither  of  the  two  methods  of  computation 
is  theoretically  tenable. 

Recognizing  that  population  is  not  stationary,  some 
writers,  such  as  Deparcieux  in  France  and  Price  in  Eng- 
land, have  proposed  that  the  total  population  be  divided 
by  the  mean  of  births  and  deaths.  Malthus  and  Charles 
Dupin  have  accepted  this  method,  and  Wappaus  too  has 
recommended  it.  Other  authors  have  computed  the  aver- 
age length  of  life  as  the  arithmetic  mean  of  the  two  ratios : 
(1)  population  to  births  and  (2)  population  to  deaths. 
Both  methods  are  purely  empirical  but  lead  to  more  accu- 
rate results  than  does  the  mere  consideration  of  either  the 
births  or  the  deaths. 

Statisticians  have  ceased  to  compute  the  average  length 
of  life  as  an  isolated  average.  Instead,  they  secure  data 
of  the  length  of  the  lives  of  the  inhabitants  of  different 
countries  and  compute  the  average  length  of  life  as  the 
arithmetic  mean  of  the  series  thus  obtained  by  observation. 
Other  averages  of  these  series  may  also  be  obtained  which 
furnish  further  standards  of  vitality.  At  first,  opinions 
differed  as  to  which  series  of  single  observations  should  be 
taken  to  compute  the  correct  average  length  of  life.     As 


ISOLATED  AVERAGES  43 

is  well  known,  the  arithmetic  average  age  of  those  living 
and  also  the  arithmetic  average  age  of  those  deceased  used 
to  be  regarded  as  the  average  length  of  life.  At  present 
the  average  length  of  life  or  expectation  of  life  at  birth  is 
computed  as  the  arithmetic  average  of  all  the  ages  in  the 
mortality  table.  A  mortality  table  represents  the  gradual 
dying  at  various  ages  of  a  number  of  people  bom  at  the 
same  time.  It  may  be  based  upon  a  concrete  observed  total- 
ity or,  as  is  the  practice,  upon  an  ideal  or  hypothetical  total- 
ity, diminished  as  the  ages  of  the  mortality  table  increase  in 
accordance  with  the  various  probabilities  of  death  which 
are  separately  determined  for  the  different  age  classes.^' 

It  is  manifest  that  the  average  age  of  those  living  at  a 
definite  time  (for  instance,  at  a  census)  does  not  correspond 
to  the  age  reached,  on  the  average,  by  the  population.  But 
the  expectation  of  life  computed  from  the  mortality  table 
also  differs  very  considerably  from  the  average  age  of  those 
who  have  died.  A  mortality  table  constructed  in  the  usual 
way  for  an  ideal  totality  from  recent  data  pictures  present 
experience.  Likewise  a  mortality  table  constructed  by  ob- 
servation of  a  concrete  group  would  express  the  mortality 
experience  of  a  definite  historical  period.  Such  is  not  the 
case  with  the  age  composition  of  the  deceased  and  with  the 
average  age  computed  from  it.  Those  who  have  died  dur- 
ing the  same  period  were  bom  in  different  years.  Their 
age  composition  is,  therefore,  caused  by  the  conditions  of 
health  which  have  prevailed  at  various  times,  and  in  com- 
puting their  average  age  we  combine  the  effects  of  inde- 
pendent causes  operating  at  different  times.    It  is  now,  in 

*''  Analogous  to  these  two  methods  of  measuring  mortality  are 
the  methods  of  measuring  the  growth  of  human  beings.  Either  a 
certain  number  of  individuals  of  the  same  age  are  observed  con- 
tinuously and  measured  periodically,  or  else  a  number  of  individuals 
of  different  ages  are  measured  at  the  same  time.  The  latter  method 
is  the  more  practicable,  just  as  it  is  easier  to  construct  a  mortality 
table  for  an  ideal  totality. 


44  STATISTICAL  AVERAGES  IN  GENERAL 

fact,  generally  agreed  that  only  the  arithmetic  average 
computed  from  the  mortality  table  is  to  be  designated  as 
the  average  length  of  life,  though,  to  be  sure,  there  are 
still  some  open  questions  as  to  the  way  a  correct  mortality 
table  should  be  computed.  Especially  the  question  of  just 
what  groups  of  living  and  deceased  we  should  take  in  de- 
termining the  correct  probabilities  of  death  is  one  of  the 
most  difficult  but  also  one  of  the  most  thoroughly  discussed 
chapters  of  statistics. 

The  method  of  computing  the  average  length  of  marriage 
has  undergone  a  similar  evolution.  Formerly  it  was  com- 
puted by  dividing  the  number  of  existing  married  couples 
by  the  annual  number  of  marriages  contracted  or  dissolu- 
tions of  marriage.  Given  an  equal  and  constant  number 
of  annual  marriages  and  dissolutions,  the  product  of  this 
number  and  of  the  average  length  of  marriage  (expressed 
in  years)  would  be  the  number  of  existing  married  couples; 
or,  the  average  length  of  marriage  would  be  the  ratio  of 
the  number  of  existing  married  couples  to  the  annual  num- 
ber of  marriages  or  dissolutions.  But  the  hypothesis  of 
an  equal  and  constant  annual  number  of  marriages  and  dis- 
solutions does  not  hold,  since  the  population  is  not  sta- 
tionary. In  an  increasing  population  the  number  of  mar- 
riages grows  more  quickly,  that  of  dissolutions  more  slowly, 
than  the  number  of  married  couples.  In  such  a  case,  there- 
fore, a  division  by  the  number  of  marriages  would  make 
the  average  length  of  marriage  too  short  and  a  division  by 
the  number  of  dissolutions  would  make  it  too  long.  Accord- 
ingly, several  statisticians  have  suggested  dividing  the  num- 
ber of  existing  married  couples  by  the  mean  of  marriages 
and  dissolutions.  Other  authors  have  divided  the  number 
of  married  couples  separately  by  the  marriages  and  (disso- 
lutions and  taken  the  mean  of  the  two  quotients.  Engel,  in 
his  work  upon  The  Movement  of  Population  in  Saxony, 
1834-1850,  used  only  the  annual  number  of  marriages  as  a 
divisor,  but  corrected  it  by  a  coefficient  computed  for  a 


ISOLATED  AVERAGES  45 

long  period  from  the  mean  annual  fluctuation  of  marriage 
frequency. 

Of  course,  it  was  soon  realized  that  these  methods  were 
only  makeshifts,  and  that  statistics  should  seek  to  gain  data 
about  the  length  of  all  individual  marriages  and  represent 
them  in  a  series.  This  has  recently  been  done  in  several 
countries.  The  length  of  all  marriages  dissolved  by  death 
or  divorce  has  been  noted  and  the  average  length  of  mar- 
riage has  been  computed  directly  from  such  data  just  as 
the  average  age  is  computed  from  the  age  classification  of 
those  who  have  died  during  a  definite  period.  The  same 
objections  may  be  made  to  the  former  computations  as  to 
the  latter.  In  computing  the  average  length  of  the  mar- 
riages dissolved  during  a  definite  period,  we  combine  mar- 
riages dating  from  different  times,  which  have  been  ex- 
posed to  various  and  independent  influences.  But  it  is 
important  to  investigate  the  length  of  the  marriages  be- 
longing to  the  same  period,  which  have  been  affected  by 
more  or  less  similar  influences.  The  most  interesting  period 
in  this  connection  is,  naturally,  the  present.  The  attempt 
has  been  made,  by  comparing  the  number  of  couples  who 
have  been  married  a  certain  length  of  time  with  those  of 
the  same  length  which  have  been  dissolved,  to  compute  prob- 
abilities of  dissolution  and  thereby  to  construct  a  marriage 
table,  analogous  to  the  mortality  tables,  for  a  period  as 
close  as  possible  to  the  present.  Such  a  table  represents 
the  progressive  diminution  from  year  to  year  of  a  hypo- 
thetical group  of  marriages  contracted  at  the  same  time. 
From  it  the  average  length  of  marriage  may  be  computed 
as  the  arithmetic  mean.  Other  averages  may  also  be  de- 
termined from  the  table,  as  the  occasion  may  require.^^ 

"  Compare  R.  Boeckh's  tables  of  the  duration  of  marriages  in 
Berlin  for  the  years  1875-6  and  1885-G,  in  the  census  report  for 
1875  (Vol.  Ill,  p.  69),  or  in  Bewegung  der  Bevolkerung  der  Stadt 
Berlin,  1869-1878  (Berlin,  1884,  p.  78),  and  in  the  Statistisches 
Jahrbuch    der    Stadt   Berlin    (Vol.    XIV,    1889,    p.    30    ff.).      From 


46  STATISTICAL  AVERAGES  IN  GENERAL 

Wappaus  and,  subsequently,  Haushofer  have  advocated  a 
different  method  of  computing  the  average  length  of  mar- 
riage. These  two  writers  think  it  may  be  obtained  by  de- 
ducting the  average  at  marriage  of  the  two  sexes  from 
their  average  length  of  life.  In  this  way  the  average  length 
of  marriage  is  not  computed  from  a  series  of  items;  and 
yet  the  computation  starts  from  two  values  which  are 
properly  averages  of  series  of  single  observations.  This 
method  is,  however,  imperfect,  because  it  considers  only 
the  marriages  dissolved  by  death,  not  those  dissolved  by 
separation.  Moreover,  not  the  general  mean  length  of  life 
but  the  special  mean  for  those  who  are  married  should  be 
used.  Even  then  the  result  would  not  be  satisfactory,  since 
what  we  desire  is  not  so  much  a  single  general  figure  for 
the  mean  length  of  marriage  as  values  for  different 
categories  of  the  population,  especially  age  classes. 

Besides  the  length  of  life  and  the  length  of  marriage, 
numerous  other  phenomena  and  conditions  are  measured 
with  regard  to  their  duration.  But  certain  difficulties 
always  attend  the  determination  of  the  average  duration 
of  mass  phenomena,  primarily  because  the  length  of  the 
phenomena  cannot  be  ascertained  by  a  single  count  or 
census.  Only  those  cases  which  exist  at  the  time  of  the 
census  are  considered,  and  even  they  are  not  characterized 

these  tables  the  average  length  of  newly  contracted  marriages  and 
of  those  of  a  certain  duration  could  be  computed.  In  order  to 
be  able  to  construct  such  tables  it  is  of  course  necessary  to  find 
out  not  only  the  length  of  all  marriages  which  have  been  terminated 
but  also  of  all  which  still  persist. 

Similarly  Boeckh  has  also  constructed  a  "marriage  table,"  based 
on  probabilities  of  marriage,  from  which  an  average  age  of  mar- 
riage may  be  computed,  which  diflTers  theoretically  from  the  kind  usu- 
ally obtained  from  the  age  distribution  of  those  marrying  in  a  certain 
period  of  time.  (Compare  Statistisches  Jahrbuch  der  Stadt  Berlin, 
1884,  p.  14,  and  Die  Bevolkerungs-  und  Wohnungsaufnahme  in  der 
Stadt  Berlin  for  December  1st,  1880,  Vol.  Ill,  §  2  B,  p.  10  S.,  Ber- 
lin, 1888.) 


ISOLATED  AVERAGES  47 

in  the  way  to  be  desired.  Only  the  duration  of  the  cases 
up  to  the  time  of  the  census  is  given;  no  conclusion  can 
be  made  about  future  duration.  Thus  a  census  furnishes 
merely  a  somewhat  unimportant  lower  limit.  For  that 
reason  neither  the  average  length  of  life  nor  the  average 
length  of  marriage  can  be  computed  from  the  data  of  the 
census.  For  the  same  reason  a  census  of  the  unemployed, 
in  which  the  latter  are  asked  about  the  length  of  time  they 
have  been  out  of  work,  does  not  give  a  theoretically  satis- 
factory result.  The  same  objection  holds  when  attempts 
are  made,  as  in  Belgium,  to  get  the  length  of  life  of 
industrial  enterprises  from  information  concerning  the 
date  of  their  establishment. 

In  order  to  be  able  to  determine  the  duration  of  mass 
phenomena,  the  constant  observation  of  the  ending  or  dis- 
appearance of  them  is  necessary.  But  constant  notations 
are  statistically  much  less  practicable  than  a  single  census. 
Hence  only  a  few  states  or  cities  note  the  length  of  dis- 
solved marriages;  data  are  lacking  about  the  duration  of 
various  diseases;  general  figures  about  the  length  of  un- 
employment are  wanting;  and  there  is  no  trustworthy  in- 
formation about  the  duration  of  industries,  buildings,  etc.^^ 

Averages  may  be  computed,  indeed,  directly  from  the  sta- 
tistical series  obtained  from  constant  notations  concerning 
the  extinction  of  the  cases  in  question.  But  we  have  just 
shown  in  connection  with  the  length  of  life  or  of  marriage 
that  averages  computed  from  series  of  immediate  observa- 
tions are,  theoretically,  not  entirely  unobjectionable.  Such 
averages  are  valid  neither  for  the  present  nor  for  a  definite 

"  Compare  Mischler's  Handbuch  der  Verwaltungsstatistik,  Vol.  I, 
§31,  "Die  Dauer  als  Eigenscliaft  der  Massenerscheinungen " ;  and 
Mischler,  "  Das  Moment  der  Zeit  in  der  Verwaltungsstatistik,"  in 
V.  Mayr's  Arehiv,  Vol.  I,  Pt.  I.  The  statute  of  the  French  Office  du 
travail  assigns  as  one  of  its  tasks  that  of  determining  the  "  dur^e 
moyenne  de  I'activite  de  I'ouvrier  dans  chaque  profession,"  a  task 
which  is  hardly  soluble,  at  least  in  France,  with  the  available  sta- 
tistical data. 


48  STATISTICAL  AVERAGES  IN  GENERAL 

past  time.  Cases  which  belong  to  different  periods  are 
grouped  together  simply  because  they  end  at  the  same  time. 
The  same  objections  may  also  be  raised  against  any  series 
which  represents  the  duration  of  phenomena  ending  at  the 
same  time,  provided  those  phenomena  extend  over  a  consid- 
erable period,  during  which  the  causes  affecting  them  may 
have  changed. 

Theoretically,  the  best  way  of  dealing  with  such  phenom- 
ena is  to  find  the  relation  between  the  existing  and  com- 
pleted cases  of  equal  duration  and  thus  to  obtain  proba- 
bilities by  means  of  which  a  table  may  be  constructed. 
From  this  table  the  average  duration  of  the  phenomena  may 
be  computed  and  other  averages  obtained.  This  method 
has,  as  we  have  indicated,  actually  been  applied  in  con- 
nection with  the  length  of  life  and  the  length  of  marriage. 
But  suggestions  only  have  been  made  in  other  directions. 
Professor  Mischler,^"  for  instance,  speaks  of  a  table  of  de- 
struction of  buildings,  whose  elements  would  be  formed 
from  the  numbers  of  existing  and  destroyed  buildings  and 
the  proofs  of  their  ages.  He  points  out  that  such  a  table 
would  form  the  only  correct  basis  for  insurance  premiums, 
rents,  the  measurement  of  the  periods  of  exemption  from 
taxation,  etc. 

Interesting  discussions  have  arisen  about  the  length  of  a 
generation.  G.  von  Mayr  defines  it  as  *'  the  average  inter- 
val between  successive  births  of  the  same  stock.  "*^  Here 
the  average  duration  of  definite  phenomena  or  conditions 
is  not  involved,  but  the  average  period  elapsing  between 
certain  events.  The  measurement  of  this  period  is  of  im- 
portance in  several  ways.  We  may  determine  by  compar- 
ing the  length  of  a  generation  and  the  length  of  life  what 
time  two  or  three  successive  generations  live  together.  *  *  In 
general,  the  whole  possibility  of  progressive  civilization  de- 
pends upon  the  relative  time  which  the  different  generations 

*"  Handbuch  der  Verwaltungsstatistik,  Vol.  I,  p.  99. 
"  Bevolkerungsstatistik,  p.  413. 


ISOLATED  AVERAGES  49 

live  together  both  within  and   without  the  lines  of  de- 
scent."" 

In  economics  the  length  of  a  generation  has  been  used  by 
Foville  to  compute  national  wealth  on  the  basis  of  inherit- 
ance taxes. 

There  are  various  methods  of  finding  the  length  of  a 
generation.  Eiimelin  claimed  that  he  obtained  it  by  add- 
ing to  the  average  age  of  men  at  marriage  half  the  marital 
period  of  fertility.*^  Obviously  this  method  leads  only  to 
an  "  isolated  "  average.  Von  Inama-Sternegg  has  sug- 
gested computing  the  length  of  a  generation  from  mass 
observations  based  on  genealogies,  and  he  has  himself  ap- 
plied this  method  to  a  considerable  extent.**  Finally, 
Vacher  and  Turquan,  taking  the  data  in  regard  to  the 
age  of  parents  furnished  by  the  birth  certificates  in  France, 
have  computed  the  length  of  a  generation  as  the  average 
age  of  the  parents  of  children  born  during  a  certain 
period.*^"*^ 

These  methods  might  also  be  judged  from  the  standpoint 
assumed  previously  in  the  criticism  of  the  methods  of 
measuring  the  length  of  life  and  of  marriage.  They  do 
not,  in  fact,  give  a  magnitude  which  refers  to  a  definite 
period;  on  the  contrary,  they  include  cases  which  belong 

**  Georg  V.  Mayr,  loc.  cit. 

**  ijber  den  Begriff  und  die  Dauer  einer  Generation,  Reden  und 
Aufsatze,  Tubingen,  1875,  p.  285  ff.  This  method,  with  a  slight 
modification,  was  also  employed  by  Goehlert;  cf.  "Die  Generations- 
dauer  vom  statistischen  Standpunkte  betrachtet,"  Stat.  Monatsschrift 
(Vienna),  Vol.  VII,  1881,  p.  49  ff. 

**"Uber  Generationsdauer  und  Generationswechsel "  (Comptes- 
rendus  et  memoires,  VIII.  Congres  international  d'Hygiene  et  de 
Demographic,   1894,  Vol.  VII,  Budapest). 

*°  Cf.  §  95  in  G.  v.  Mayr's  Bevolkerungsstatistik,  "  Die  Genera- 
tionen,"  and  also  the  literature  mentioned  there. 

*°Quetelet  seems  to  have  understood  (wrongly)  length  of  genera- 
tion to  mean  the  average  age  of  men  or  women  at  the  birth  of  their 
first  child  (cf.  Versueh  einer  Physik  der  Gesellschaft,  German  edi- 
tion, 1838,  p.  66  at  top). 


50  STATISTICAL  AVERAGES  IN  GENERAL 

to  different  periods.  The  question  might,  therefore,  well 
be  asked,  how  the  precise  length  of  a  generation  could  be 
determined  for  the  present  or  for  some  definite  historical 
period. 

The  method  of  measuring  marital  fecundity  offers  in  its 
historical  development  another  interesting  example  of  the 
progress  from  an  isolated  mean  to  one  obtained  from  a 
series  of  observations.  It  is,  at  the  same  time,  an  example 
of  the  difficulty  of  choosing,  from  several  series  which 
apparently  represent  the  phenomenon,  that  one  which  will 
furnish  the  correct  mean. 

Marital  fecundity  may  be  viewed  from  two  different 
standpoints.  The  annual  frequency  of  legitimate  births 
may  be  investigated.  For  this  purpose  the  legitimate  births 
of  a  year  are  usually  placed  in  relation  to  the  married 
women  living  at  the  same  time,  either  with  the  total 
number  or  only  with  those  of  child-bearing  age.  If  greater 
accuracy  is  desired,  the  infants  may  be  divided  into  groups 
according  to  the  age  of  their  mothers  and  may  be  placed 
in  relation  to  the  numbers  of  the  married  women  in  the 
age  classes  in  question.  Furthermore,  the  infants  may  be 
compared  also  with  the  numbers  of  married  men,  and 
finally  the  various  numbers  of  infants,  differentiated  ac- 
cording to  the  ages  of  their  parents,  may  be  compared  with 
the  marriages  of  corresponding  age  combinations.  It  was 
in  this  latter  way  that  Korosi  obtained  his  "  Bigenous 
Table  of  Natality,"  which  represents  the  frequency  of 
births  for  the  various  age  combinations  of  parents  and 
which  was  submitted  to  the  Royal  Society  of  London  in 
1893,  just  200  years  after  Halley  had  prepared  the  first 
mortality  table  and  presented  it  to  the  same  society.*^ 

*^  See  "An  Estimate  of  the  Degrees  of  Legitimate  Natality  as 
derived  from  a  Table  of  Natality  compiled  by  the  Author  from  his 
Observations  made  at  Budapest"  in  Vol.  CLXXXVI,  2.  1895  B.  of 
the  Philosophical  Transactions  of  the  Royal  Society  of  London, 
pp.  781-875.     Korosi  published  a  short  summary  of  this  article  in 


ISOLATED  AVERAGES  51 

It  is  a  different  problem,  however,  to  determine  how 
many  children  on  an  average  result  from  a  marriage  during 
its  entire  length  (or  how  many  are  to  be  expected),  in 
which  case,  of  course,  special  averages  for  marriages  of 
different  lengths  and  for  different  age  combinations  of  the 
parents  are  to  be  sought.  Korosi,  in  the  paper  cited 
above,  distinguishes  therefore  between  the  measurement  of 
fecundity  in  general  and  the  measurement  of  "  richness  of 
marriages  "  or  "  expectation  of  children."  His  distinc- 
tion is  of  fundamental  importance.  In  the  following  pages 
we  deal  only  with  the  methods  by  which  marital  fecundity 
to  death  or  separation  (that  is,  the  average  number  of 
children  for  the  entire  length  of  marriage)  is  determined. 

Until  fairly  recently  it  was  the  custom  to  obtain  the 
life  fecundity  by  dividing  the  number  of  births  during  a 
year  by  the  number  of  marriages  or  by  the  number  of 
dissolutions  for  the  same  year.  Korosi  quotes  this  as  a 
method  of  obtaining  marital  fertility  in  the  narrower 
sense.  Yet,  as  a  matter  of  fact,  this  method,  though  de- 
pending upon  a  comparison  of  annual  figures,  may  under 
definite  circumstances  give  the  number  of  births  which 
occur  on  an  average  during  the  whole  length  of  marriage. 

Lexis  makes  the  following  statement  of  the  case:  if  b 
represents  the  total  number  of  legitimate  births  and  m 
the  total  number  of  marriages  which  occur  in  a  generation 
of  females,  then  evidently  the  ratio  —  expresses  the 
measure  of  life  fecundity.*^  In  a  stationary  population  we 
could  replace  b  and  m  (which  are  not  available)  by  the 
number  of  legitimate  births  and  the  number  of  marriages 
in  a  definite  year  and  the  ratio  of  the  two  numbers  would 
express  the  fecundity.     But  this  is  true  only  if  the  popu- 

1895  in  the  Revue  d'Sconomie  politique  (Paris)  under  the  title 
"  De  la  mesure  et  des  lois  de  la  fgcondite  conjugale."  Korosi's  Table 
of  Natality  was  adjusted  by  Galton  and  Dr.  Blaschke  ( Vienna ) . 

**  Cf.  Abhandlungen  zur  Theorie  der  Bevolkerungs-  und  Moral- 
statistik,  IV,  "  Ubersicht  der  demographischen  Elemente  und  ihrer 
Beziehungen  zu  Einander,"  p.  81. 


52  STATISTICAL  AVERAGES  IN  GENERAL 

lation  is  stationary.  Even  Siissmilch  and  Malthus  pointed 
out  this  fact,  as  Korosi  has  mentioned  in  his  paper  "  Zur 
Erweiterung  der  Natalitats-  und  Fruchtbarkeitsstatistik. ' '  ** 
Even  if  the  population  were  stationary,  this  method  of  com- 
puting marital  fecundity  would  be  imperfect,  because  in- 
stead of  making  a  distinction  between  sterile  and  fruitful 
marriages  it  gives  a  common  average  for  these  two  essen- 
tially different  categories. 

But,  as  is  well  known,  population  is  not  stationary. 
Therefore,  if  the  measure  of  fertility  is  computed  according 
to  the  method  in  question,  the  result  will  depend  chiefly 
on  the  number  of  marriages  or  dissolutions  during  the 
given  year.  The  mere  decrease  of  marriages  or  dissolutions 
will  result  in  an  increase  in  the  quotient,  indicating  appar- 
ently a  greater  fertility ;  and  conversely,  a  smaller  measure 
of  fecundity  will  result  from  an  increase  of  marriages  and 
dissolutions. 

Such  a  method  is,  therefore,  extremely  dubious.  More- 
over, it  is  not  made  correct  by  dividing  the  births  by  the 
arithmetic  mean  of  the  marriages  and  dissolutions  of  the 
same  year.  Nevertheless,  this  latter  modification  of  the 
method  has  prevailed  until  very  recently.  Haushofer  as 
late  as  1882  advocated  it  in  his  LeJir-  und  Handbuch  der 
Statistik  (p.  407). 

Two  other  improvements  of  the  method  have  been  at- 
tempted. First,  the  ratio  between  births  and  marriages  has 
been  determined  for  a  longer  period  than  a  year,  so  that 
a  larger  percentage  of  the  births  might  come  from  the 
marriages  used  in  the  comparison  than  would  be  the  case 
for  a  single  year.  Secondly,  not  the  marriages  of  the 
same  year   (or  period)   as  the  births  are  used,  but  those 

"  Bulletin  de  I'lnst.  intern,  de  Stat.,  Vol.  VI,  2nd  issue,  Supple- 
ment 22  f.  p.  b. ;  for  further  discussions  about  the  measurement  of 
fertility  and  the  use  of  it  by  various  authors,  see  the  same  refer- 
ence, and  also  R.  Boeckh's  "  Die  statistische  Messung  der  ehelichen 
Fruchtbarkeit "  in  the  Bulletin,  Vol.  V,  Ist  issue,  p.  159  S. 


ISOLATED  AVERAGES  53 

of  an  earlier  period,  which  precedes  the  period  of  the 
births  by  the  average  interval  between  marriage  and  the 
birth  of  a  child.  Dr.  Farr  has  computed  this  interval  to 
be  six  years,  and  has  accordingly  divided  the  number  of 
births  by  the  number  of  marriages  contracted  six  years 
before.  The  divisor  is,  therefore,  somewhat  smaller  than 
when  the  marriages  of  the  same  year  are  taken,  where 
the  numer  of  marriages  is  increasing,  and  thus  the  marital 
fecundity  appears  greater.  Wappaus  has  employed  the 
tedious  method  of  comparing  the  arithmetic  mean  of  the 
annual  legitimate  births  for  three  years  with  the  mean 
of  the  annually  newly  contracted  marriages  (and,  if  pos- 
sible, also  of  the  dissolutions)  of  the  seven  preceding 
years.^" 

Modern  statisticians  try  to  obtain  the  average  marital 
fecundity  not  as  an  isolated  quotient  but  as  an  average 
from  series  of  observations  of  the  number  of  children  in 
individual  marriages.  This  detailed  observation  renders 
possible  the  distinction  between  sterile  and  fruitful  mar- 
riages. Of  fruitful  marriages  those  of  different  lengths 
and  different  age  relations  of  the  parents  may  be  differen- 
tiated and  special  averages  for  the  number  of  children  in 
these  categories  may  be  computed. 

The  first  attempts  to  compute  the  average  number  of 
children  per  marriage  on  the  basis  of  detailed  observations 
utilized  the  results  of  the  census,  in  so  far  as  questions 
had  been  asked  about  the  number  of  children  bom  in 
wedlock  and  about  the  length  of  the  marriages.  But  it 
is  evident  that  such  data  do  not  give  a  measure  of  the 
full  marital  fecundity.  The  marriages,  which  the  census 
ascertains,  persist  and  in  many  of  them  additional  chil- 
dren are  born.  Such  data  signify  merely  a  minimum,  and 
give  no  information  about  the  number  of  children  born 
during  the  whole  length  of  marriage.^^     Boeckh  has,  how- 

'"  Allgem.  Bevolkerungsstatistik,  Pt.  II,  p.  314. 

"*  See  the  states  or  cities,  for  which  there  are  data  concerning 


54  STATISTICAL  AVERAGES  IN  GENERAL 

ever,  attempted  to  compute  marital  fecundity  on  the  basis 
of  the  census  data  and  of  the  table  of  lengths  of  marriage, 
presuming  that  the  dissolved  marriages  of  a  definite  dura- 
tion might  as  a  rule  have  the  same  numbers  of  children  as 
the  persisting  marriages  of  the  same  duration,  which  be- 
long to  approximately  the  same  period  of  fertility.^^ 

Another  method  of  measuring  marital  fecundity  is  to 
number  each  birth  according  to  its  order  in  the  family  in 
question,  and,  if  possible,  also  at  the  same  time  to  ascertain 
the  age  of  the  parents  and  the  length  of  the  marriage. 
But  this  method  is  as  unsatisfactory  as  the  one  previously 
described.  The  marriages  investigated  are  not  yet  termi- 
nated and  other  children  may  be  born.  Furthermore,  the 
percentage  of  sterile  marriages  cannot  be  determined  from 
such  lists.^^  In  spite  of  these  difficulties,  Boeckh  in  his 
Berlin  statistics  undertook,  by  an  ingenious  working  over 
of  the  birth  data  and  with  the  aid  of  the  mortality  table, 
to  deduce  the  fecundity,  just  as  he  had  attempted  it  on 
the  basis  of  the  census  data.  The  results  obtained  in  this 
way  differed  considerably  from  those  obtained  directly 
from  the  data  of  births.^* 

Only  very  recently  have  the  marriages  dissolved  by 
death  or  separation  been  observed  with  reference  to  the 

existing  marriages  according  to  the  number  of  children,  in  A.  N. 
Kiaer's  Statistische  Beitrlige  zur  Beleuchtung  der  ehe'ichen  Frucht- 
barkeit,  Pt.  Ill,  Christiania,  1905   (Tables  1,  3,  and  4). 

»^  See  Die  Berliner  Volksziihlung  von  1885,  Vol.  II,  Pt.  II,  p.  50  ff. 
March  has  tried  to  utilize  the  results  of  the  French  census  of  1901 
to  determine  fecundity  by  considering  only  marriages  of  longer  dura- 
tion (more  than  15  years,  20  years).  Compare  his  Families  Parisi- 
ennes,  Composition-Fecondite. 

"  See  the  states  or  cities  for  which  we  have  data  of  the  births 
according  to  their  ordinal  number  in  Kiaer's  Statistische  Beitrage 
zur  Beleuchtung  der  ehelichen  Fruchtbarkeit,  Christiania,  1905, 
Pt.  Ill  (survey  in  Table  2). 

°*  The  results  of  this  computation  for  the  years  1886-1890  and 
1891-1895  are  given  in  the  Statistisches  Jahrbuch  der  Stadt  Berlin 
fUr  das  Jahr  1899,  p.  104. 


ISOLATED  AVERAGES  55 

number  of  children  bom  from  them  in  order  to  compute 
the  average  number  of  children  per  marriage  from  the 
statistical  series  thus  obtained.  This  method  requires  that 
we  should  ascertain,  in  the  case  of  all  deaths  of  married 
persons  and  in  the  case  of  all  separations,  the  number  of 
children  bom  from  the  marriages.  But  since  fecundity 
varies  according  to  length  of  marriage  and  according  to 
the  age  of  the  parents,  it  is  also  desirable  that  these  latter 
facts  should  be  ascertained." 

But  even  this  method  of  computing  marital  fecundity  is 
not  entirely  unobjectionable.  The  same  criticisms  may  be 
made  against  it  that  were  made  against  the  computation 
of  the  average  length  of  life  from  the  age  classification 
of  the  deceased  or  against  the  computation  of  the  average 
length  of  marriage  from  the  classification  of  dissolved 
marriages  according  to  their  length.  It  might  be  shown 
that  the  marriages  terminated  during  a  definite  period 
(say,  a  year)  had  been  contracted  during  the  preceding 
decades  and  were  therefore  subject  to  varying  influences. 
The  mean  computed  for  such  marriages  is,  accordingly, 
not  suitable  for  characterizing  the  fecundity  of  the  pres- 
ent time;  indeed,  it  does  not  apply  to  any  definite  period. 
Professor  Zoltan  Rath  has  formulated  this  objection  in  his 
"  Memoir e  sur  la  methode  la  plus  simple  de  mesurer  la 
fecondite  des  manages."  ^^  He  says:  "  Although  the  sta- 
tistics of  marriages  dissolved  by  death  include  also  newly 
contracted  marriages,  yet  the  majority  of  the  marriages 
whose  fecundity  is  examined  date  some  time  back,  since 
in  the  nature  of  things  the  dissolution  of  marriages  often 
comes  after  several  decades  of  duration.  The  method  in 
question,  therefore,  represents  the  demographic  habits  of 

°°  Korosi,  for  example,  has  obtained  such  exhaustive  data  for 
Budapest.  Cf.  his  "  Weitere  Beitrage  zur  Statistik  der  ehelichen 
Fruchtbarkeit "  in  the  Bulletin  de  I'Inst.  intern,  de  Stat.,  Vol.  XIII, 
Pt.  III. 

=°  Bulletin  de  I'Inst.  intern,  de  Stat.,  Vol.  XIII,  Pt.  II. 


56  STATISTICAL  AVERAGES  IN  GENERAL 

past  generations  rather  than  of  the  present."  But  Profes- 
sor Rath  is  inclined  not  to  lay  any  great  stress  on  this  ob- 
jection. He  says  later  on:  '*  While  admitting  that  death 
frequently  happens  only  after  a  period  already  sterile  in 
the  lives  of  married  women,  it  may  be  asserted  that  in 
the  majority  of  marriages  so  dissolved  the  period  of  fecun- 
dity is  not  past.  The  greater  the  general  mortality,  the 
nearer  our  method  will  follow  the  fecundity  of  the  present 
generation." 

At  all  events,  the  question  arises,  in  what  way  the  precise 
average  fecundity  of  the  present  may  be  computed.  The 
way  seems  to  be  indicated  by  the  somewhat  analogous  de- 
velopment of  the  method  of  computing  the  average  length 
of  life.  It  was  once  thought  that  the  average  length  of 
life  might  be  found  in  the  average  age  of  the  deceased. 
Now  it  is  computed  from  a  mortality  table  based  on  the 
probabilities  of  death.  Might  not  the  average  marital 
fecundity  be  similarly  computed  from  probabilities  of 
birth?  Ludwig  Moser  seems  to  have  thought  of  this.  In 
his  work  Die  Gesetze  der  Lehensdauer  (1839),  which 
was  epoch-making  in  its  day,  he  said  that  in  order  to 
determine  fecundity  we  must  make  our  observations  in 
connection  with  the  ages  of  the  parents  so  as  to  ascertain 
what  percentage  of  married  women  at  the  age  of  30  have 
a  child,  what  at  the  age  of  31,  etc. ;  the  sum  of  these  would 
give  the  marital  fecundity.^'' 

G.  von  Mayr  appears  to  have  a  similar  method  in  mind. 
He  asserts  in  his  BevolJcerungsstatistik  (p.  185)  that 
it  would  be  possible  to  obtain  **  a  complete  picture  of 
marital  fecundity  in  its  gradation  according  to  the  age 
conditions  of  the  parents  combined  with  length  of  mar- 
riage (fecundity  table)  "  as  follows  :^^    "  1.  Directly  and 

"^  See  Boeckh,  "  Die  statistische  Messung  der  ehelichen  Frucht- 
barkeit,"  Bulletin  de  I'Inst.  intern,  de  Stat.,  Vol.  V,  Pt.  I,  p.  162. 

°'  V.  Mayr's  Table  of  Fecundity  is  quite  different  from  Korosi's 
Natality  Table.     The  latter  gives  only  special  birth  rates    (accord- 


ISOLATED  AVERAGES  57 

strictly  historically,  by  taking  a  certain  group  of  mar- 
riages (say,  those  of  a  year)  and  dividing  them  according  to 
the  ages  of  the  husbands  and  wives  and  then  ascertaining 
for  each  marriage  when  dissolved  the  number  of  children 
classified  according  to  the  duration  of  the  married  life. 
When  the  last  marriage  of  the  group  is  dissolved,  the 
fecundity  of  the  various  groups  of  marriages,  arranged 
according  to  age  combinations  and  lengths,  is  obtained. 
2.  Indirectly,  and  ideally,  by  combining  the  simultaneous 
experiences  of  various  groups.  A  single  group  of  mar- 
riages is  not  observed  through  the  different  years  of  its 
duration,  but  fragments  of  short  observations  of  marriages 
of  different  lengths  are  used  to  obtain  a  theoretical  fecun- 
dity for  an  ideal  group.  It  is,  of  course,  necessary  that 
we  know  the  age  conditions  and  the  length  of  the  existing 
marriages  and  that  the  same  facts  are  known  for  every 
birth.  If  we  have  all  these  data  the  fecundity  of  all  kinds 
of  marriages  may  be  computed,  especially  such  as  last 
until  the  procreative  period  ceases."  The  result  obtained 
by  the  first  method  of  Mayr  would  refer  to  a  generation 
already  past,  in  which  there  would  probably  be  but  little 
interest.  His  second  method,  on  the  other  hand,  would 
be  analogous  to  the  present  prevailing  method  of  com- 
puting the  average  length  of  life,  that  is,  it  would  be  based 
on  probabilities  determined  for  the  present.  The  latter 
method,  however,  has  one  great  fault.  It  does  not  express 
the  important  difference  between  fruitful  and  sterile  mar- 
riages. Moreover,  both  methods,  so  far  as  they  demand  a 
consideration  of  the  length  of  marriage  and  of  the  age 
conditions  of  parents,  presuppose  data  wMch  are  not  ob- 
tained at  present.^^ 

ing  to  the  age  classes  of  the  parents),  where^^s  Mayr's  Table  is 
intended  to  represent  the  development  of  fecundity  in  an  historical 
or  ideal  totality  of  marriages  from  contraction  to  dissolution. 

"  The   International    Statistical    Institute   in   its    10th    and    11th 
sessions    (1905,   1907)    took  up  the  question  of  the  formulation  of 


58  STATISTICAL  AVERAGES  IN  GENERAL 

Korosi  is  indeed  of  the  opinion  that  marital  fecundity 
cannot  be  computed  at  all  on  the  basis  of  probabilities  of 
birth.  In  this  connection  he  says  in  his  paper  already  re- 
ferred to,  *'  An  Estimate  of  the  Degrees  of  Legitimate  Na- 
tality," etc.  (p.  867  f.)  :  "  One  might  perhaps  think 
that  the  addition  of  the  natalities  stated  for  each 
year  of  the  procreative  period  would  furnish  the 
probability  for  this  whole  period.  To  prove  the  im- 
practicability of  such  a  proposition  it  is  sufficient  to 
point  out  the  physiological  fact  that  female  con- 
ception stops  not  only  during  childbed,  but  even  during 
the  period  of  lactation.  There  exists  between  two  births 
a  natural  interval,  which,  moreover,  is  further  increased 
by  the  moral  moment.  .  .  .  The  idea  that  a  wife  dur- 
ing five  years  from  the  age  of  30  to  that  of  35  could 
undergo  individually  the  birth  probabilities  obtained  for 
the  total  of  the  wives  at  the  age  of  30,  31,  32,  33,  and  34 
years,  is  wrong;  to  observe  for  one  year  the  natality  of 
five  mothers,  each  of  them  being  one  year  older  than  the 
other,  and  to  observe  the  natality  of  one  for  five  subsequent 
years, — these  are  two  different  things."  Korosi 's  argu- 
ments, however,  do  not  appear  cogent.®"  The  single  birth 
probabilities  refer  to  definitely  characterized  groups  of  in- 
dividuals; similarly  marital  fecundity  is  to  be  determined 
for  groups  of  homogeneous  marriages.  It  may  be  imagined, 
therefore,  that  a  definitely  characterized  group  of  marriages 
would  be  subject  successively  to  the  birth  probabilities  ob- 

statisties  of  fecundity  and  formed  a  number  of  conclusions  based  on 
reports  of  Korosi,  March,  and  Kiaer  (cf.  Bulletin,  Vol.  XV,  Pt.  II, 
and  Vol.  XVII). 

'°  Korosi's  "Natalities"  would  of  course  be  insuflSeient,  since  they 
are  concerned  only  with  the  age  relationships  of  the  parents  but  not 
with  the  length  of  marriage.  Besides,  as  v.  Bortkiewicz  has  shown 
(cf.  his  discussion  of  Korosi's  Table  of  Natality  in  the  Jahrbiicher 
flir  Nationalokonomie  und  Statistik,  1897,  No.  1,  p.  123  ff.),  these  are 
not  really  numerical  probabilities  but  "  intensity  coefficients  of 
marital  fertility  for  individual  age  classes." 


ISOLATED  AVERAGES  59 

tained  for  the  present  time  for  the  age  relations  and  length 
of  marriage  in  question;  and  accordingly  the  total  and 
average  number  of  children  might  be  computed  up  to  the 
dissolution  of  marriage  on  the  basis  of  these  probabilities. 
The  birth  probabilities  also  take  account  of  the  fact  men- 
tioned by  Korosi  that  "  female  conception  stops  not  only 
during  childbed,  but  even  during  the  period  of  lactation," 
since  the  women  who  are  temporarily  incapable  of  con- 
ception are  contained  in  the  denominator  of  the  probability 
fractions,  whence  follows  a  corresponding  decrease  in  the 
quotient. 


CHAPTER  III 

NECESSITY  OF  THE  LOGICAL  AGREEMENT  OF  MAGNI- 
TUDES FROM  WHICH  AN  AVERAGE  IS  TO  BE  COM- 
PUTED 

Statistical  series,  from  which  averages  are  computed, 
consist  either  of  quantitative  single  observations  or  of 
values  which  characterize  masses  limited  in  a  definite  way 
in  regard  to  their  absolute  size  or  in  other  respects  (by 
relative  numbers  or  averages). 

If  a  series  consists  of  quantitative  single  observations, 
these  must  agree  as  regards  both  the  observation  unit  and 
the  observation  element  in  order  to  produce  a  clear  and 
precisely  definable  average.  If,  for  instance,  the  wages  in 
a  definite  occupation  are  to  be  represented  in  the  form  of 
an  average,  then  only  those  laborers  should  be  considered 
who  belong  to  that  occupation,  and  the  element  of  measure- 
ment, "  wages,"  must  be  conceived  and  obtained  in  the 
same  manner  in  connection  with  all  the  laborers.  The  lat- 
ter would  not  be  the  case,  if,  for  instance,  both  money 
wages  and  wages  in  kind  were  considered  in  the  case  of 
some  laborers  and  merely  money  wages  in  the  case  of 
others.  The  average  wage  computed  from  such  a  series 
would  be  neither  the  average  money  wage  nor  the  average 
total  wage ;  indeed,  it  could  not  be  defined  exactly. 

If  a  series  does  not  consist  of  single  observations,  but 
either  of  values  which  indicate  the  size  of  definitely  limited 
masses  (series  of  the  second  group)  or  of  values  which 
characterize  such  masses  in  other  ways  by  relative  numbers 
or  averages,  then  the  individual  values  do  not  agree  com- 
pletely, but  differ  from  each  other  according  to  a  criterion 

60 


LOGICAL  AGREEMENT  OF  ITEMS  61 

of  time,  place,  quantity,  or  quality.  But  this  is  the  only 
difference  which  may  exist  between  the  individual  values, 
so  that  these  values  must  be  like  several  species  of  the  same 
genus.  When  the  average  is  computed,  the  criterion  used 
to  differentiate  the  items  is  disregarded,  so  that  these  items 
agree  completely  at  that  moment.  If,  for  instance,  we 
have  the  numbers  of  births  for  a  series  of  years,  these 
values  are,  to  be  sure,  differentiated  in  regard  to  time, 
but  must  agree  in  all  other  respects,  which  would  not  be 
the  case  if,  for  example,  in  some  years  all  the  births,  in 
other  years  only  the  living  births,  were  taken.  In  com- 
puting the  average  number  of  births  per  year  the  time- 
difference  of  the  items  is  ignored  and  the  mean  size  of 
masses,  each  corresponding  to  the  same  generic  conception, 
is  computed.  Relative  numbers,  especially,  which  refer 
to  definitely  limited  masses,  must  agree  exactly  in  the  way 
in  which  they  characterize  the  masses  to  which  they  refer. 
It  would  accordingly  be  incorrect  to  obtain  an  average 
from  birth  rates  which  included  the  stillborn  for  some 
years  but  not  for  others. 

With  certain  reservations,  therefore,  the  necessity  of 
a  logical  agreement  of  the  magnitudes  from  which  an  aver- 
age is  to  be  computed  may  be  postulated.  In  the  case  of 
single  observations  this  necessity  is  absolute.  In  series 
of  the  second  and  third  groups  the  items  must  correspond 
to  the  same  generic  conception,  and  therefore  must  logically 
agree,  at  least  at  the  moment  of  computing  the  mean,  at 
which  time  the  criterion  used  to  differentiate  the  parts  is 
ignored. 

This  logical  agreement  of  magnitudes  from  which  an 
average  is  to  be  computed  is  by  no  means  always  satisfied. 
The  agreement  of  the  items  of  a  series  is  not  always  easy 
to  establish,  especially  in  time  and  geographical  series 
where  the  members  originate  from  distinct  investigations. 
In  different  returns  the  same  object  may  easily  be  differ- 
ently defined  and  limited.     If  in  two  successive  censuses 


62  STATISTICAL  AVERAGES  IN  GENERAL 

the  idea  of  a  "  household  "  were  differently  conceived,  it 
would  hardly  be  permissible  to  compute  from  these  two 
censuses  an  average  of  "  households  ";  such  an  average 
could  not  be  clearly  defined.  Similarly  the  annual  sums 
of  exports  and  imports  would  not  be  logically  corresponding 
magnitudes,  if  at  different  times  varying  quantities  of 
wares  were  excepted,  or  if  the  methods  of  determining  the 
values  had  changed  considerably.  In  series  obtained  from 
a  single  investigation  it  is  of  course  assumed  that  the  con- 
ception of  the  object  does  not  vary.  Mistaken  interpreta- 
tions may  be  regarded  as  accidental  errors,  which  do  not 
invalidate  the  logical  unity  of  the  series. 

The  rule  that  averages  must  be  obtained  from  logically 
agreeing  magnitudes  was  formerly  frequently  violated. 
We  shall  not  speak  of  errors  due  to  extreme  carelessness. 
Those  cases  deserve  mention,  however,  in  which  magnitudes 
of  different  kinds  were  consciously  but  mistakenly  em- 
ployed to  compute  averages.  Thus,  as  has  already  been 
mentioned,  the  attempt  used  to  be  made  to  obtain  the 
average  length  of  life  either  by  dividing  the  population  on 
the  one  hand  by  the  number  of  births  and  on  the  other  hand 
by  the  number  of  deaths  and  then  computing  an  average 
from  the  two  manifestly  unlike  quotients,  or  by  dividing 
the  population  directly  by  an  average  of  births  and  deaths 
— an  average  of  two  entirely  heterogeneous  magnitudes. 
In  the  same  way  it  used  to  be  thought  that  the  average 
length  of  marriage  was  obtained  by  dividing  the  number 
of  existing  marriages  on  the  one  hand  by  the  dissolved, 
and  on  the  other  hand  by  the  newly  contracted  marriages, 
and  determining  the  average  of  the  two  quotients,  or  else 
by  dividing  the  number  of  existing  marriages  by  the  aver- 
age of  the  dissolved  and  the  newly  contracted  marriages. 
Such  computations  had  no  scientific  basis ;  they  were  mere 
makeshifts.  Two  possibilities  were  seen  of  computing  the 
average  length  of  life ;  in  both  methods  the  dividend  was 
the   same    (namely,    the    population),    while    there    were 


LOGICAL  AGREEMENT  OF  ITEMS  63 

two  divisors,  the  number  of  births  and  the  number  of 
deaths.  It  was  found  that  by  using  one  divisor  too  large 
a  figure  was  obtained,  by  using  the  other,  too  small  a 
figure.  Accordingly,  the  simple  device  was  adopted  of 
taking  the  average  of  the  quotients  or  of  the  divisors. 

Formerly,  on  account  of  the  undeveloped  condition  of  sta- 
tistics, it  happened  frequently  that  several  methods  would 
be  employed,  not  one  of  which  was  felt  to  be  perfectly 
reliable,  and  then  an  average  would  be  taken  of  the  results 
of  the  different  methods.  The  political  arithmeticians 
often  did  this;  Petty,  for  example,  computed  the  period 
of  doubling  of  the  population  by  different  methods  and 
then  took  an  average  of  his  results.^^  Lajinen  often  have 
recourse  to  these  makeshifts.  Leibnitz  relates  in  his 
Nouveaux  Essais  sur  I'entendement  humain  (1700) 
that  in  Lower  Saxony  when  a  piece  of  property  was  to  be 
sold  the  peasants  used  to  form  three  groups,  each  of  which 
made  an  estimate,  and  then  the  average  of  the  three  esti- 
mates was  fixed  as  the  price.®^  Often,  when  it  was  dis- 
covered that  mortality  tables  computed  in  different  ways 
gave  different  results,  averages  were  taken  from  several 
tables.  "Westergaard  remarks  in  this  connection:  ^^  *'  Such 
an  employment  of  several  tables,  as  if  they  were  all 
equally  good  observations,  is  now  rightly  regarded  as  ir- 
rational. ' ' 

This  process  of  taking  an  average  of  results  obtained  in 
different  ways  reminds  one  in  certain  respects  of  the  re- 
peated observations  of  the  same  object,  so  common  in 
astronomy  and  geodesy,  and  of  the  "  objective  "  means 
computed  from  those  observations.  In  neither  case  is  it 
a  question  of  obtaining  an  average  for  a  series  of  con- 

"  Cf.  Westergaard,  Die  Grundziige  der  Theorie  der  Statistik,  p. 
255. 

"  Mentioned  in  the  Journal  of  the  Royal  Statistical  Society, 
Vol.  LIV,  1891,  p.  451. 

*'  Die  Lehre  von  der  Mortalitat  und  Morbilitat,  p.  106. 


64  STATISTICAL  AVERAGES  IN  GENERAL 

Crete  plienomena  of  different  sizes,  but  of  the  correct 
ascertainment  of  a  single  fact,  of  the  most  probable  value 
of  a  magnitude,  for  which  by  reason  of  insufficient  methods 
of  observation  several  values  are  suggested.  However, 
there  is  an  important  difference.  In  astronomy  and  geodesy 
we  have  to  deal  with  manifold  measurements,  undertaken 
according  to  the  same  method  and  with  the  same  instru- 
ments, subject  only  to  accidental  errors  which  are  to  be 
eliminated  as  far  as  possible  by  the  computation  of  an 
average.  On  the  other  hand,  when,  for  example,  an  aver- 
age is  computed  from  several  mortality  tables,  we  are  treat- 
ing as  equivalent  results  obtained  by  means  of  different 
methods  and  therefore  showing  definite  variations.  But 
it  is  the  duty  of  science  in  such  a  case  to  ascertain  the 
most  correct  method  and  to  accept  the  result  of  it;  it  is 
an  abdication  of  the  scientific  method  to  regard  results  of 
different  methods  as  of  equal  value  and  to  blend  them  in 
an  average. 


CHAPTER  IV 

POSTULATE  OF  THE  GREATEST  POSSIBLE  HOMO- 
GENEITY OF  SERIES  FROM  WHICH  AVERAGES 
ARE  COMPUTED,  AND  OF  MASSES  WHICH  ARE 
CHARACTERIZED  BY  RELATIVE  NUMBERS 

The  significance  of  averages  consists  in  the  fact  that  they 
express  the  result  of  the  activity  of  definite  complexes  of 
causes  in  one  characteristic  figure.  Thus,  the  average  wage 
of  a  definite  group  of  laborers  gives  a  measure  of  the  factors 
determining  the  level  of  wages  in  that  group.  Now  it 
is  of  great  importance  that  the  average  shall  refer  to  a 
complex  of  causes  as  nearly  unified  as  possible,  since  only 
in  this  way  will  it  possess  a  definitely  intelligible  content, 
and  only  in  this  way  may  reliable  inferences  be  drawn 
from  a  change  in  the  average.®*  If  masses  of  items,  which 
have  evidently  been  variously  influenced  by  quite  inde- 
pendent causes,  are  taken  together  in  a  series  the  average 
so  computed  has  little  scientific  value,  since  it  does  not 
express  the  activity  of  a  unified  complex  of  natural  or 
social  causes  and  is,  as  a  rule,  poorly  adapted  to  purposes 
of  comparison.  If  the  wages  in  two  different  branches  of 
industry  are  determined  by  quite  different  causes,  then  the 
average  wages  for  all  the  laborers  in  both  industries  cannot 
be  regarded  as  a  measure  of  the  factors  operative  in  either 
of  them,  and  hence  no  trustworthy  inference  may  be  dra^^Ti 
from  a  change  in  the  average.  For  these  reasons  modern 
statisticians  try  to  form  series  of  individual  values  as 
nearly  homogeneous  as  possible  and  to  compute  averages 
only  from  such  series.     This  is  true  both  of  series  of  single 

•*  Cf.  below,  p.  101  f. 

65 


66  STATISTICAL  AVERAGES  IN  GENERAL 

observations  and  also  of  series  whose  members  indicate 
the  magnitude  of  definitely  limited  masses  (parts  of  a 
larger  totality).  For  similar  reasons,  in  computing  rela- 
tive numbers,  the  general  object  is  to  distinguish  masses 
as  nearly  homogeneous  as  possible,  and  to  characterize 
them  by  special  relative  numbers. 


A.  POSTULATE  OF  THE  GREATEST  POSSIBLE  HOMO- 
GENEITY OF  SINGLE  OBSERVATIONS  FROM  WHICH 
AN  AVERAGE  IS  COMPUTED 

There  are  two  problems  to  be  distinguished  in  following 
the  tendency  to  compute  averages  from  series  of  single 
observations  as  nearly  homogeneous  as  possible.  The  first 
problem  is  to  eliminate  such  individual  cases  as  do  not  show 
the  observation  element  in  question;  the  second  problem  is 
to  divide  into  masses  as  nearly  homogeneous  as  possible 
the  individual  cases  which  actually  do  show  the  observa- 
tion element. 

First  problem.  In  the  statistical  observation  of  a  number 
of  items  it  is  often  evident  that  a  part  of  them  do  not 
show  the  measurement  or  other  observation  element  at 
all.  The  question  now  arises,  whether  in  such  circum- 
stances the  whole  series  is  to  be  considered  in  computing 
the  average  or  only  those  cases  where  observation  has 
yielded  a  positive  result.  This  question  is,  indeed,  unimpor- 
tant for  the  determination  of  the  mode,  but  may  seriously 
affect  the  arithmetic  mean  or  the  median. 

The  question  cannot  be  answered  in  general.  In  those 
items  which  do  not  show  the  observation  element,  the  causal 
complex  which  produces  it  in  other  items  may  not  have 
been  operative  at  all.  In  such  a  case  the  inclusion  of  the 
items  not  showing  the  observation  element  would  influence 
the  result  wrongly.  The  average  size  of  the  observation 
element  would  then  appear  smaller  than  would  be  the  case 
if  only  positive  results  were  taken;  or,  in  other  words,  it 


HOMOGENEITY  OF  SERIES  67 

would  depend  essentially  on  the  proportion  of  observations 
with  and  without  a  positive  value.  On  the  other  hand, 
if  there  is  no  essential  difference  in  the  causation  of  the 
two  classes  of  observations  there  is  no  reason  to  eliminate 
the  latter. 

The  elimination  of  the  items  which  do  not  show  the 
observation  element  is  often  a  matter  of  course,  but  in 
other  cases  the  decision  of  the  question  is  very  difficult. 
In  computing  average  wages,  for  example,  cases  where 
laborers  for  personal  reasons  receive  no  wages  at  all  will 
naturally  be  disregarded.  More  difficult,  on  the  other 
hand,  is  the  question  of  dealing  with  sterile  marriages 
when  the  average  number  of  children  per  marriage  is  to 
be  computed.  Generally,  a  common  average  for  fruitful 
and  unfruitful  marriages  is  computed.  But  more  recently 
the  demand  has  arisen  to  disregard  sterile  marriages  com- 
pletely and  to  consider  only  those  marriages  in  which  there 
is  at  least  one  child.  Of  course,  in  that  case  the  average 
will  be  considerably  higher.  The  reason  alleged  is  that 
absolute  unfruitfulness  of  a  marriage  is  normally  a  patho- 
logical phenomenon  and  is  to  be  attributed  to  disability 
or  sickness  in  one  or  both  of  the  parents.  The  causes 
influencing  the  degree  of  fecundity,  it  is  said,  are  not 
operative  in  such  unfruitful  marriages,  and  a  correct  ex- 
pression of  these  causes  can  only  be  obtained  by  eliminating 
wholly  unfruitful  marriages  in  the  computation  of  an 
average.  But  these  arguments  are  not  convincing.  Not 
all  unfruitful  marriages  should  be  attributed  to  patholog- 
ical causes.  It  is  probable  that  "  neo-Malthusianism," 
which  generally  affects  the  degree  of  fecundity  by  limiting 
the  number  of  children,  sometimes  leads  to  complete  aban- 
donment of  reproduction.  In  that  case,  neo-Malthusianism 
would  represent  a  cause  affecting  both  fertile  and  sterile 
marriages.  The  same  is  also  true  of  diseases  which  affect 
reproduction.  If  such  a  disease  is  present  at  the  outset, 
the  marriage  will  remain  unfruitful;   but  if  it  appears 


68  STATISTICAL  AVERAGES  IN  GENERAL 

during  the  marriage  after  children  have  already  been 
bom,  it  will  affect  merely  the  number  of  children  or  the 
degree  of  fecundity  of  the  marriage.  It  cannot  be  asserted, 
therefore,  that  the  causes  of  unfruitful  marriages  are  quite 
independent  of  the  causes  which  determine  the  degree  of 
fecundity  of  fruitful  marriages.  However,  a  separate  treat- 
ment of  unfruitful  marriages  is  no  doubt  necessary.  The 
percentage  of  unfruitful  marriages  is  of  the  greatest  in- 
terest. Average  marital  fecundity  is  to  be  given,  if  pos- 
sible, both  inclusive  and  exclusive  of  them.*'^"^'^ 

Of  coui-se,  to  disregard  unfruitful  marriages  must  be 
considered  theoretically  as  a  makeshift.  The  aim  is  to 
keep  apart  items  influenced  by  different  causes.  As  a  mat- 
ter of  fact,  the  items  in  question  are  differentiated  accord- 
ing to  whether  a  definite  effect  has  been  produced  or  not. 
The  more  correct  proceeding  (although  of  course  not  fea- 
sible in  the  present  instance)  would  be  to  differentiate  the 
items  according  to  the  criterion  to  which  the  non-appearance 
of  the  observation  element  is  attributable.  If,  then,  we 
assume  that  unfruitful  marriages  are,  as  a  rule,  to  be 
attributed  to  pathological  causes,  we  should  not  eliminate 
the   unfruitful  marriages  but  those  marriages   in  which 

''  Dr.  Friedrich  Prinzing  distinguishes  in  his  Handbuch  der 
medizinischen  Statistik  (p.  31)  between  childless  marriages  in 
which  no  child  capable  of  living  is  born  and  sterile  marriage  in 
which  not  even  a  miscarriage  has  taken  place.  Practical  statistics 
cannot  make  this  fine  distinction  but  regards  all  marriages  as  child- 
less or  sterile  in  which  there  is  no  offspring  either  living  or  still- 
born. Besides  Prinzing,  A.  N.  Kiaer  (in  Pts.  I  and  II  of  his 
Statistische  Beitrage  zur  Beleuchtimg  der  eheliehen  Fruchtbarkeit, 
Christiania,  1903)  has  offered  an  exhaustive  comparative  presenta- 
tion of  childless  marriages. 

"'  For  similar  reasons  it  is  also  desirable  that  the  average  length 
of  life  should  be  computed  both  for  all  births  (including  stillborn) 
and  separately  for  those  born  alive.  According  to  the  German  mor- 
tality table  the  average  length  of  life  for  the  latter  is  35.58,  for 
the  former  34.04  years. 


HOMOGENEITY  OF  SERIES  69 

pathological  causes  occur,  and  thus  the  homogeneity  of  the 
series  would  be  established. 

The  elimination  of  those  items  which  do  not  show  the 
observation  element  is,  of  course,  only  possible  on  condition 
that  individual  observations  have  been  made.  But  since 
such  individual  observations  are  not  made,  for  instance, 
in  regard  to  the  consumption  of  alcohol  or  meat  for  the 
whole  population,  manifestly  those  persons  or  classes  who 
do  not  consume  alcohol  or  meat  cannot  be  eliminated.  The 
average  consumption  of  alcohol  or  meat  per  head  for  the 
entire,  population  may  indeed  be  computed  as  an  "  isolated 
average,"  but  no  average  which  applies  only  to  the  con- 
sumers can  be  obtained,  and  it  remains  unkno\\'n  whether 
the  whole  population  or  only  a  fraction  of  it  takes  part  in 
the  consumption  in  question. 

General  averages  of  the  kind  just  indicated  do,  however, 
possess  a  certain  scientific  value,  even  when  it  is  possible 
to  compute  special  averages  for  the  classes  of  individuals 
actually  concerned.  Such  averages  give  a  measure  of  the 
significance  which  the  phenomena  possess  for  a  -u-ider 
though  indirectly  interested  circle  of  people.  In  fact  it 
is  customary  in  different  branches  of  statistics,  even  where 
individual  data  are  at  hand,  to  compute  not  only  special 
averages  for  those  personally  concerned,  but  also  general 
averages  for  a  wider  circle,  including  also  those  indirectly 
interested.  Thus,  in  connection  with  sick  funds,  not  only 
the  average  number  of  days  per  case  of  sickness  is  com- 
puted, but  also  the  average  number  of  days  for  each  mem- 
ber of  the  organization,  in  which  case  account  is  taken  of 
members  who  have  not  been  sick  at  all,  since  they  too  are 
indirectly  affected  by  the  duration  of  the  illnesses,  the 
amount  of  their  contributions  depending  upon  it.  Sim- 
ilarly, both  the  average  taxes  per  taxpayer  and  per  head 
of  the  population  are  computed,  the  latter,  so  to  speak,  as 
a  measure  of  the  burden  on  the  whole  population.  In 
these   and  similar  cases   the   absence   of  the   element   of 


70  STATISTICAL  AVERAGES  IN  GENERAL 

measurement  (length  of  sickness,  amount  of  taxes)  must 
not  be  attributed  to  radical  differences  in  causation.  The 
same  causes  which  diminish  the  size  of  the  element  of 
measurement  may  by  operating  more  intensively  cause 
it  to  disappear  entirely.  Hence  it  is  permissible  and, 
from  a  certain  point  of  view,  instructive  to  take  together 
all  the  items,  including  those  which  do' not  show  the  element 
of  observation. 

Second  problem.  The  endeavor  to  obtain  homogeneous 
series  of  single  observations  is  only  partly  satisfied  by 
eliminating  cases  which  do  not  show  the  element  of  ob- 
servation. Among  those  cases  which  have  yielded  positive 
data,  special  parts  may  often  be  distinguished  which  are 
influenced  by  different  and  independent  causes.  The  statis- 
tician must  try  in  such  cases  to  divide  the  whole  series 
into  more  homogeneous  parts.  If  this  is  not  done,  the 
average  computed  from  the  whole  series  is  not  the  expres- 
sion of  a  unified  complex  of  causes,  and  so  does  not  allow 
any  reliable  inferences  to  be  based  upon  it.  Its  magnitude 
will  depend  chiefly  on  the  proportion  in  which  the  various 
more  homogeneous  parts  forming  the  series  stand  to  one 
another.  For  instance,  let  us  suppose  that  a  series  of 
wage  data  includes  the  wages  of  all  laborers  in  a  district. 
Now  it  is  well  known  that  sex  has  a  determining  influence 
on  wages.  Women  generally  receive  lower  wages.  We 
have,  therefore,  first  of  all  to  separate  men  and  women. 
If  this  is  not  done,  the  value  of  the  average  wages  of  men 
and  women  together  will  depend  on  the  proportion  in  which 
the  two  sexes  are  represented,  without  furnishing  any  in- 
formation about  the  wage  conditions  of  either  sex  by 
itself. 

But  sex  is  by  no  means  the  only  factor  influencing 
wages  which  statistics  can  determine.  Laborers  must  also 
be  distinguished  in  regard  to  occupation,  category  of  work, 
age,  etc.  In  this  way  the  statistician  will  form  more 
homogeneous   groups    and   compute    special    averages   for 


HOMOGENEITY  OF  SERIES  71 

them.  It  should  also  be  mentioned  here,  that  series  of 
heterogeneous  parts  generally  show  an  irregular  formation 
— often  with  several  points  of  concentration — but  that  by- 
disintegrating  such  series  regular  constituent  series  are 
often  obtained  which  furnish  a  "  typical  "  mean. 

In  fact,  in  all  branches  of  statistics  there  is  a  noteworthy 
tendency  to  form  more  detailed  homogeneous  masses.  Not 
only  wages  but  also  various  other  observations  are  differ- 
entiated according  to  sex,  conjugal  condition  and  age, 
wherever  feasible.  Distinctions  of  occupation,  economic 
condition,  etc.,  are  sought.  Still  further  divisions  may  be 
suggested  by  the  nature  of  the  investigation;  for  example, 
in  connection  with  marital  fertility,  marriages  may  be 
divided  according  to  their  length  and  to  the  age  of  the 
contracting  parties.®^ 

There  are,  however,  many  difficulties  to  be  overcome  in 
forming  homogeneous  masses.  In  particular,  the  differen- 
tiation of  the  mean  length  of  life  for  various  classes  of 
the  population  (according  to  occupation,  wealth,  dwelling 
conditions,  etc.)  is  still  very  incomplete;  and  the  same  is 
true  of  averages  representing  age  at  marriage,  marital 
fecundity,     etc.     Many    series    cannot    be     divided     into 

"  The  principle  of  forming  the  most  homogeneous  groups  possible 
is  also  to  be  applied  to  estimates.  In  Austria  the  political  authori- 
ties ascertain  the  daily  wages  in  the  various  judicial  districts  of  all 
workmen  who  come  under  the  sickness-insurance  law.  "  If  con- 
siderable divergencies  are  shown,  the  ordinary  wages  may  be  ex- 
pressed in  several  categories.  Separate  statements  are  made  for 
male,  female,  juvenile,  and  adult  laborers.  Apprentices,  assistants, 
unsalaried  clerks,  and  others  who  draw  small  wages  or  none  at 
all  are  classified  among  the  juvenile  laborers"  (§7,  Kranken- 
versicherungsgesetz ) .  The  decree  of  the  Ministry  of  the  Interior  of 
January  20,  1894,  says,  moreover,  that  in  case  these  distinctions  are 
insufficient,  other  categories  may  be  formed,  especially  of  male  la- 
borers drawing  full  wages — for  instance,  into  foremen,  artisans, 
factory  employees,  and  ordinary  day  laborers.  Furthermore  it  may 
be  often  necessary  to  make  distinctions  among  the  various  groups 
of  industries. 


72  STATISTICAL  AVERAGES  IN  GENERAL 

homogeneous  groups  at  all,  although  their  structure  shows 
that  they  are  made  up  of  heterogeneous  components. 

The  postulate  of  homogeneity  is  not  limited  to  criteria 
of  quantity  and  quality.  Homogeneous  groups  of  observa- 
tions are  also  to  be  sought  in  connection  with  criteria  of 
time  and  place.  It  is  a  manifest  disadvantage  that  an 
average  should  be  computed  from  items  belonging  to  widely 
different  periods.  Of  particular  significance  is  the  differ- 
entiation according  to  abstract  time  criteria ;  the  most  im- 
portant of  these  are  the  seasons,  which,  as  is  well  known, 
exert  a  considerable  influence  on  demographic  and  economic 
phenomena.  It  is  also  undesirable  to  treat  very  large 
geographical  districts  as  units.  The  average  in  this  way 
loses  reality,  and  becomes  a  mere  abstraction.  Here  too  a 
differentiation  according  to  abstract  geographic  criteria  is 
possible — altitude,  soil,  climate,  etc.  Especially  a  differen- 
tiation of  the  length  of  life  according  to  such  criteria  is 
much  sought  after. 

B.  POSTULATE  OF  THE  GREATEST  POSSIBLE  HOMO- 
GENEITY IN  THE  COMPUTATION  OF  AN  AVERAGE 
FROM  VALUES  WHICH  EXPRESS  THE  SIZES  OF 
MASSES  LIMITED  IN  A  DEFINITE  WAY  (CONSTITU- 
ENTS OF  A  GREATER  TOTALITY) 

Series  of  the  second  group  which  do  not  consist  of  single 
observations  but  w^hich  express  the  size  of  definitely  limited 
masses,  may  also  be  tested  in  regard  to  the  homogeneity 
of  the  items.  Time  series  of  absolute  figures  are  especially 
to  be  considered  here.  Just  as  there  are  single  observa- 
tions with  the  numerical  value  of  zero,  so  the  item  zero 
may  also  occur  in  a  time  series.  Such  a  member  may  be 
disregarded  in  the  computation  of  the  average  if  it  was 
subject  to  abnormal  time  influences.  But  such  cases  occur 
only  rarely.  It  will  frequently  be  possible,  on  the  other 
hand,  to  keep  apart  periods  of  time  in  which  essentially 
different  causes  were  operative,  and  to  characterize  them 


HOMOGENEITY  OF  SERIES  73 

by  special  averages.  Accordingly,  separate  averages  ought, 
if  possible,  to  be  computed  for  the  years  preceding  a 
new  cause  and  for  those  which  follow. 

When  an  average  is  computed  from  time  series,  the 
maximum  and  minimum  of  the  series  are  often  disregarded. 
This  is  done  from  an  intuitive  effort  after  homogeneity. 
It  is  assumed  that  the  extreme  cases  have  been  influenced 
by  special,  transitory  causes,  and  that  accordingly  such 
abnormal  cases  should  not  be  considered  if  an  average  is 
to  be  obtained  representing  probable  future  development. 
But  the  maximum  and  minimum  need  not  always  be  ab- 
normal eases  arising  from  exceptional  conditions.  And, 
on  the  other  hand,  there  may  be  other  abnormal  cases  be- 
sides the  maximum  and  minimum.  Therefore,  when  an 
average  is  computed  from  a  time  series,  it  is  only  per- 
missible to  disregard  certain  years  (no  matter  how  or  to 
what  extent  they  differ  from  the  average)  when  they  are 
demonstrably  subject  to  exceptional  causes  which  supplant 
the  normal  causes.®^ 

C.  POSTULATE  OF  THE  GREATEST  POSSIBLE  HOMO- 
GENEITY OF  IMASSES  WHICH  ARE  CHARACTERIZED 
BY  RELATIVE  NUMBERS 

Relative  numbers,  which  are  to  be  regarded  as  averages, 
do  not  arise  by  computation  from  series  of  individual  values, 
but  are  obtained  by  independent  subdivision  or  coordination 
of  statistical  masses.  Therefore,  the  postulate  that  only 
homogeneous  individual  values  should  be  considered  must 
be  somewhat  modified  in  connection  with  relative  numbers. 
The  demand  should  be  made  here  that  masses  as  homo- 

'*  Single  years  may  also  be  disregarded  for  definite  non-statistical 
reasons.  Thus  various  Austrian  railroad  concessions  state  that  in 
order  to  determine  the  purchase  price  the  seven  years  of  operation 
preceding  the  purchase  are  taken  and  of  these  the  two  most  unfa- 
vorable years  are  eliminated  and  then  the  average  net  profits  for  the 
remaining  five  years  are  computed. 


74  STATISTICAL  AVERAGES  IN  GENERAL 

geneous  as  possible  be  distinguished  and  characterized  by 
special  relative  numbers  (subordinate  or  coordinate  num- 
bers). Accordingly,  masses  not  participating  in  the  phe- 
nomenon in  question  are  often  eliminated,  and  even  masses 
with  positive  measurements  are  generally  divided  into  more 
homogeneous  parts. 

Elimination  of  non-participating  masses  in  the  computa- 
tion of  relative  numbers.  In  computing  subordinate  num- 
bers it  may  be  necessary  to  eliminate  masses  which  by  their 
very  nature  belong  to  one  definite  subdivision  of  the  whole 
and  cannot  belong  to  any  other.  Thus,  for  instance,  chil- 
dren are  only  unmarried,  never  married  or  separated. 
Hence  the  division  of  the  whole  population  according  to 
family  condition  is  of  doubtful  value;  the  ratio  of  the 
married  to  the  unmarried  depends  essentially  on  the  age 
division  of  the  population;  a  large  number  of  children 
naturally  increases  the  percentage  of  the  unmarried.  It 
is,  therefore,  more  to  the  purpose  to  eliminate  that  portion 
of  the  population  not  yet  capable  of  marriage. 

Of  greater  importance  is  the  elimination  of  non-partici- 
pating masses  in  computing  various  demographic  coordi- 
nate numbers.  These  are  often  computed  by  relating  defi- 
nite events  and  the  entire  population.  But  in  the  whole 
population  constituent  masses  may  often  be  distinguished 
in  which  such  events  cannot  occur.  Thus  it  is  impossible 
for  children  to  contribute  to  the  number  of  births  or  mar- 
riages. If  a  measure  of  the  causes  determining  the  fre- 
quency of  marriage  or  birth  is  to  be  obtained,  it  is  neces- 
sary to  eliminate  from  the  divisor  those  classes  in  which 
these  causes  are  not  operative.  This  is  done  by  relating 
the  births  only  to  those  age  classes  of  the  population 
capable  of  reproduction,  the  marriages  only  to  people  of 
marriageable  age.  In  this  way  not  general  but  specific 
figures  are  secured. 

Yet  some  significance  must  be  conceded  to  the  general 
figures,  since  they  represent  the  phenomenon  in  question 


HOMOGENEITY  OF  SERIES  75 

from  the  standpoint  of  the  total  population  who  are  at 
least  indirectly  interested.  G.  von  Mayr  has  defended 
general  frequency  figures  from  this  point  of  view.^^  He 
notes  that  they  are  often  looked  upon  with  scorn,  and 
goes  on  to  say :  ' '  This  scorn  is  justified,  so  far  as  we  have 
in  mind  the  question  of  the  subjective  participation  of 
the  population  in  the  events.  But  it  is  not  justified  if 
we  observe  further  that  not  only  the  factor  of  subjective 
participation  or  responsibility  is  important,  but  also  the 
objective  burden  (in  the  good  and  bad  senses)  of  the 
whole  population.  Crimes  have  a  statistical  interest  not 
only  from  the  standpoint  of  the  subjective  participation  of 
individual  classes  of  the  population,  but  also  as  an  ob- 
jective disturbance  of  the  whole  community.  Births  and 
deaths  may  also  be  so  regarded.  They  are  not  only  in- 
teresting as  indicating  the  contribution  of  the  classes  capa- 
ble of  reproduction  and  as  indicating  the  dangers  to  life 
of  the  different  classes  but  they  are  also  significant  socially 
and  economically  in  their  relation  to  the  total  popula- 
tion." 

Division  of  masses  into  more  homogeneous  parts.  The 
elimination  of  non-participating  masses  is  frequently  in- 
sufficient. Within  the  participating  masses,  constituents 
may  often  be  distinguished  in  which  a  very  definite  char- 
acter prevails,  or  in  which  the  phenomenon  appears  with 
different  intensity.  Where  this  is  to  be  attributed  to  differ- 
ent causes  influencing  the  whole  part  in  question,  it  is 
obviously  desirable  to  keep  such  part  separate  and  compute 
for  it  special  relative  numbers. 

As  an  example  of  the  disintegration  of  a  mass  into  more 
homogeneous  parts,  let  us  take  again  the  division  of  the 
population  according  to  conjugal  condition.  If  the  com- 
position of  the  whole  marriageable  population  is  repre- 
sented according  to  conjugal  condition    (single,  married, 

"  "  Die  statistischen  Gesetze."  Public  lecture  of  August  27,  1895 
(Bulletin  de  I'Inst.  intern,  de  Stat.,  Vol.  IX,  Pt.  II,  p.  304  f.). 


76  STATISTICAL  AVERAGES  IN  GENERAL 

separated,  etc.),  no  homogeneous  mass  lies  at  the  basis  of 
the  subordinate  numbers.  Such  a  division  affects  men  and 
women  differently;  it  is  well  known  that  there  are  more 
widows  than  widowers  because  of  the  different  ages  of  the 
sexes  at  marriage.  Such  a  division  also  varies  for  different 
ages.  As  recent  investigations  have  shown,  it  likewise 
varies  in  different  social  classes;  the  possibilities  of  mar- 
riage of  industrial  laborers  differ  from  those  of  the  agri- 
cultural population.  Possibilities  of  marriage  also  depend 
on  economic  conditions,  etc.  It  is,  therefore,  of  great  in- 
terest to  determine  the  division  according  to  conjugal  con- 
dition not  only  for  the  whole  marriageable  population  but 
also  for  the  more  homogeneous  parts  of  it. 

More  important  are  the  cases  in  which,  in  the  computa- 
tion of  coordinate  numbers,  the  masses  to  be  related  may 
be  divided  into  more  homogeneous  parts.  The  differentia- 
tion of  the  demographical  frequency  numbers,  such  as  birth 
rate,  death  rate,  marriage  rate,  etc.,  is  to  be  considered 
in  this  connection.  The  criteria  to  be  applied  are  the  same 
as  are  used  to  obtain  more  homogeneous  subordinate  num- 
bers or  to  divide  series  of  single  observations  into  more 
homogeneous  groups.  Sex,  age,  and  conjugal  condition 
are  most  commonly  used.  But  occupation,  dwelling,  san- 
atory conditions,  etc.,  should  also  be  employed. 

The  tendency  of  modern  statistics  is  everywhere  to  ob- 
tain relative  numbers  for  masses  as  nearly  homogeneous 
as  possible.  In  this  process,  relative  numbers  which  are 
by  nature  averages  are  often  resolved  into  more  special 
values,  which  must  indeed  still  be  designated  as  averages, 
but  which  are  related  to  the  original  relative  number  as 
individual  values  for  smaller  parts.  This  process  evidently 
increases  in  importance  with  the  increase  of  grades  of  in- 
tensity included  in  the  original  relative  number. 

Not  only  quantitative  and  qualitative  homogeneity  but 
also  unity  in  regard  to  matters  of  place  and  time  are  to 
be  sought  in  relative  numbers.    Homogeneity  in  place  leads 


HOMOGENEITY  OF  SERIES  77 

to  the  formation  of  "  natural  districts,"  which  can  be 
marked  off  according  to  topographic,  hydrographic,  and 
other  points  of  view  ("  geographic  method  "  in  von  Mayr's 
terminology  in  contradistinction  to  the  "  statistical  geo- 
graphic method,"  by  which  the  geographic  distribution  of 
the  various  grades  of  a  phenomenon  is  represented).  In 
order  to  obtain  homogeneity  in  matters  of  time,  Professor 
Mischler  has  suggested  the  formation  of  "  natural  time- 
periods,"''**  differing  from  the  usual  calendar  divisions 
and  combining  periods  Avhich  exhibit  the  same  frequency 
of  events. 


D.    SIMPLE  INDICATION  OF  THE  RANGE  OF  A 
SERIES 

In  accordance  with  the  principle  that  averages  should 
be  computed  as  far  as  possible  from  series  whose  items  may 
be  regarded  as  homogeneous,  the  average  is  often  not  found 
for  heterogeneous  series.  So,  too,  the  computation  of  a 
relative  number  as  an  average  of  certain  special  values  is 
often  abandoned,  when  this  relative  number  would  have 
to  be  computed  on  the  basis  of  very  heterogeneous  masses. 
"When  the  computation  of  the  average  is  found  inadvisable, 
there  remains  nothing  to  do  but  enumerate  all  the  indi- 
vidual values  or  form  magnitude  classes  from  them.  This, 
however,  does  not  result  in  such  simplification  and  brevity 
as  the  computation  of  an  average  would  give.  Therefore, 
the  minimum  and  maximum  of  the  series,  the  so-called 
"  range  "  of  the  series,  are  often  substituted.'^     For  in- 

''"  Handbuch  der  Verwaltungsstatistik,  Vol.  I,  p.  89 ;  cf.  also  by 
the  same  author  "  Das  Moment  der  Zeit  in  der  Verwaltungsstatistik  " 
in  V.  Mayr's  Allg.  Stat.  Archiv,  Vol.  I,  Pt.  I. 

^'  This  procedure  is  often  chosen  without  reference  to  the  homo- 
geneity of  the  series  or  of  the  masses  in  question  because  it  requires 
no  work  and  yet  is  sufficient  for  certain  purposes. 

This  method  of  characterizing  a  series  by  simply  mentioning  its 


78  STATISTICAL  AVERAGES  IN  GENERAL 

stance,  frequently  only  the  highest  and  lowest  prices  of  a 
commodity  are  given,  when  the  computation  of  an  average 
price  would  be  valueless  because  of  considerable  qualitative 
differences.  Stock  quotations  also  are  generally  given  only 
in  highest  and  lowest  prices. 

This  last  process  is  often  employed  in  order  to  express 
briefly  geographical  series  of  relative  numbers  which  refer 
to  various  countries.  Thus,  the  general  birth  rate  for 
different  countries  for  the  years  1887-1891  may  be  said  to 
have  varied  from  22.8  (Ireland)  to  42.8  (Hungary)." 
Apart  from  technical  difficulties,  it  would  hardly  be  per- 
missible to  compute  averages  from  such  geographical  series, 
since  a  wholly  untypical  value  would  result.  Even  within 
the  same  country  the  conditions  may  vary  to  such  an  ex- 
tent that  a  general  average  would  appear  useless.  In  con- 
nection with  the  agricultural  wages  in  Austria  in  1893, 
von  Inama-Sternegg  refused  to  compute  averages  for  larger 
districts  from  the  average  wages  for  the  various  court- 
districts  given  by  agricultural  experts.  "  The  value  of 
these  court-district  data  consists  in  their  reality;  averages 
for  larger  districts  or  for  whole  countries  would  obliterate 
all  characteristic  differences;  the  larger  the  territory,  the 
farther  removed  the  average  would  be  from  reality,  without 
other  compensating  advantages  aside  from  securing  a  for- 
mal, unified  expression.     On  the  other  hand,  it  seems  useful 

highest  and  lowest  values  is  to  be  distinguished  from  the  case  where 
in  securing  data  only  the  maximum  and  minimum  of  a  phenomenon 
(for  example,  the  highest  and  lowest  prices  or  wages)  are  obtained. 
Thus  the  Austrian  Labor  Statistical  oflBce,  in  securing  data  con- 
cerning the  miners  in  a  certain  district,  also  obtained  data  in  regard 
to  the  agricultural  industry  of  the  same  district.  The  wages  of  the 
agricultural  laborers  were  in  part  ascertained  only  in  maximum  and 
minimum  figures.  (Cf.  Arbeiters'erhaltnisse  im  Ostrau-Karwiner 
Steinkohlenreviere,  Dargestellt  vom  k.  k.  Arbeitsstatistischen  Amt 
im  Handelsministerium,  Pt.  I,  p.  xxv  and  p.  577  ff. ) 

'^  Cf.  the  figures  for  the  individual  countries  in  v.  Mayr's  Bevol- 
kerungsstatistik,  p.  177. 


HOMOGENEITY  OF  SERIES  79 

to  record  the  maximum  and  minimum  values  found  for 
the  larger  districts. ' '  " 

"  The  agricultural  wages  in  the  kingdoms  and  provinces  repre- 
sented in  the  Reichsrat  according  to  special  data  collected  by  the 
Ministry  of  Agriculture  for  the  year  1893.  Elaborated  by  the  bureau 
of  the  Statistical  Central  Commission  in  Vienna.  Osterreichische 
Statistik,  Vol.  XLIV,  Pt.  I,  Vienna,  1896,  p.  ix. 


CHAPTER  V 

FORMATION  OF  MAGNITUDE  CLASSES 

Statistics  can  only  rarely  measure  all  the  degrees  of  mag- 
nitude and  intensity  which  natural  and  social  phenomena 
display.  Frequently,  the  exact  measurement  of  the  items 
is  not  attempted ;  the  statistician  merely  determines  to  what 
magnitude  classes  they  belong.  Even  when  the  items  are 
measured  exactly,  the  individual  observations  are,  as  a  rule, 
finally  united  in  magnitude  classes,  and  even  those  statis- 
tical series,  which  do  not  consist  of  single  observations  but 
of  values  of  other  kinds,  are  often  treated  similarly.  This 
method  shows  certain  analogies  with  the  method  of  com- 
puting averages.  The  same  tendency  toward  simplification 
and  abbreviation  by  the  omission  of  unimportant  details 
leads  first  to  the  formation  of  magnitude  classes  and  finally 
also  to  the  computation  of  averages.  It  will,  therefore,  be 
advantageous  to  discuss  briefly  the  formation  of  magnitude 
classes  before  discussing  the  purpose  of  averages.  This 
order  is  also  necessary  because  the  statistical  series,  from 
which  an  average  is  to  be  computed,  frequently  consist 
not  of  single  observations  but  of  magnitude  classes,  and  this 
fact  must  be  especially  considered  in  the  computation  of 
the  average.'^* 

Frequently,  as  we  have  stated,  items  are  not  accurately 

'*  Cf.  for  the  questions  connected  with  magnitude  classes: — G.  v. 
Mayr,  Theoretische  Statistik,  §  43,  "  Die  Zusaramenziige,"  and  §  45, 
"  Verhaltnisberechnungen "  (p.  94);  A.  Bertillon,  "La  theorie  des 
moyennes  en  statistique  "  (Journal  de  la  Society  de  Statistique  de 
Paris,  1876,  p.  302);  G.  Th.  Fechner,  Kollektivmasslehre,  VII, 
"  Primare  Verteilungstafeln  "  (§§  47-52),  and  VIII,  "  Reduzierte  Ver- 
teilungstafeln  "   ( §§  53-67 ) . 

80 


FORMATION  OF  MAGNITUDE  CLASSES  81 

measured,  but  merely  assigned  to  a  magnitude  class.  If  a 
tabular  form  is  used  for  recording  the  observations,  the 
number  of  magnitude  classes  is  a  priori  limited  and  nothing 
is  noted  except  the  number  of  observations  falling  into  each 
class.  But  even  individual  records  do  not  always  imply 
exact  measurements.  Thus,  years  are  often  asked  for  where 
months  and  days  would  be  possible;  in  anthropometric 
measurements  one  is  frequently  satisfied  to  determine  the 
size  to  the  nearest  centimeter.  In  such  cases  individual 
differences  which  do  not  reach  a  certain  amount  are  not 
expressed,  and  items  which  apparently  coincide  might  dif- 
fer if  greater  accuracy  were  observed.  The  units  of  meas- 
urement that  are  distinguished  (in  the  above  examples,  the 
years  or  centimeters)  are  really  magnitude  classes,  and  the 
observations  are  made  with  sole  regard  to  these  classes. 

Certain  "  continuous  "  phenomena  can  never  be  meas- 
ured with  complete  accuracy,  so  that  the  resulting  data 
always  possess  the  character  of  magnitude  classes.  For 
instance,  space,  time,  and  weight  measurements  usually  con- 
cern continuous  phenomena.  If  the  phenomenon  is  discon- 
tinuous, consisting  in  the  individual  case  of  indivisible, 
concrete  units,  then  all  the  items  can  be  ascertained  with 
complete  accuracy.  But  such  accuracy  is  not,  as  a  rule, 
necessary.''^ 

The  preparation  and  publication  of  material  usually  leads 

'°  In  forming  magnitude  classes  a  distinction  must  be  made  be- 
tween continuous  and  discontinuous  data.  The  latter,  consisting  of 
units  not  further  divisible,  occur,  for  example,  where  human  beings 
of  a  definite  category  are  concerned.  Links  between  such  units 
are  inconceivable.  For  instance,  if  industries  are  classified  accord- 
ing to  the  number  of  their  laborers,  one  class  will  end  with  49 
laborers  and  the  next  begin  with  50.  On  the  other  hand,  where 
measurements  are  concerned  such  a  method  of  delimiting  the  classes 
is  impossible.  There  may  be  innumerable  links  between  49  and 
50  cm.  It  is  therefore  more  accurate  to  quote  a  single  figure  as 
a  limit,  so  that  one  class  may  reach  to  50  cm.  inclusive,  and  the 
next  begin  with  over  50. 


82  STATISTICAL  AVERAGES  IN  GENERAL 

to  still  further  leveling  of  details.  For  instance,  even  where 
accurate  age  data  are  obtained,  in  publication  only  years 
are  given,  perhaps  only  age  classes  of  several  years  each. 
The  number  of  laborers  in  industrial  enterprises,  or  the 
wages  of  laborers,  or  incomes,  or  the  like,  are  generally 
published  only  according  to  magnitude  class.  Individual 
differences,  which  are  smaller  than  the  range  of  the  mag- 
nitude classes  concerned,  cannot  then  be  determined.  Series 
of  quantitative  single  observations  consist,  accordingly, 
either  of  the  original  individual  magnitudes  expressed  with 
more  or  less  accuracy,  or  of  magnitude  classes.  The  items 
belonging  to  the  different  magnitude  classes  may  be  given 
in  absolute  numbers  or  as  percentages  of  the  total  number 
of  items.'^^ 

Similarly,  series  of  the  second  group,  namely,  those  whose 
members  indicate  the  magnitude  of  definitely  limited  masses, 
may  be  grouped  in  magnitude  classes.  Thus,  the  districts 
of  a  country  may  be  united  in  several  groups  according 
to  their  population.  The  same  is  true  of  series  of  the  third 
group,  those  which  consist  of  relative  numbers  or  averages, 
by  which  definitely  limited  masses  are  characterized  other- 
wise than  as  regards  their  magnitude.  Instead  of  giving 
the  density  of  population  or  the  average  wages  for  each 
single  district  of  a  country,  magnitude  classes  may  be 
formed  and  the  number  of  the  districts  in  each  class  given. 
Time,  place,  and  qualitative  series  are  changed  of  course, 

■'»  Fechner  (Kollektivmasslehre,  §§47-67)  divides  measurement 
data  into  the  "  first  lists,"  on  the  one  hand,  and  "  primary "  and 
"reduced  tables,"  on  the  other.  In  the  "first  list"  the  measure- 
ments are  given  in  the  accidental  order  in  which  they  were  ob- 
tained. When  these  measurements  are  arranged  according  to  size, 
a  "  primary  table "  is  effected.  By  grouping  the  measurements  a 
"reduced  table"  is  obtained.  In  practical  statistics,  however,  there 
is  no  necessity  of  constructing  a  "  primary  table "  when  magnitude 
classes  are  formed  whose  limits  are  predetermined.  In  such  cases  the 
individual  measurements  are  simply  assigned  to  the  classes,  and  do 
not  need  to  be  arranged  in  them  according  to  size. 


FORMATION  OF  MAGNITUDE  CLASSES  83 

when  magnitude  classes  are  formed,  into  quantitative 
series.'''^ 

Magnitude  classes  are  scientifically  justifiable.  By  the 
formation  of  groups  the  original  data,  often  very  extensive, 
are  compressed  into  a  series  which  can  more  easily  be  sur- 
veyed and  judged.  In  well  chosen  magnitude  classes  the 
structure  of  the  mass  finds  clear  expression ;  regularities 
are  revealed  which  might  not  otherwise  be  discovered.  It 
is  also,  naturally,  much  easier  to  compare  two  or  more 
simplified  series  of  magnitude  classes  than  series  which  give 
the  full  details  of  the  original  material.  Of  course,  in  the 
individual  case  everything  depends  on  the  kind  of  mag- 
nitude classes  formed.  The  object  should  not  be  to  form 
as  many  classes  as  possible  but  to  form  such  as  express 
what  is  characteristic  in  the  structure  of  the  mass.  Both 
the  extent  and  the  position  of  the  magnitude  classes  are 
of  consequence ;  for  magnitude  classes  of  equal  extent  may, 
according  to  their  positions,  reveal  different  characteristics. 

It  is  also  worth  noting  that  in  correctly  formed  classes 
a  large  part  of  the  errors  which  affect  individual  observa- 
tions disappear.  A  good  example  of  this  is  the  age  classi- 
fication of  the  inhabitants  of  those  countries  in  which  the 
age,  not  the  date  of  birth,  is  asked  in  the  census.  If  each 
age  year  is  given  separately,  it  will  be  found  as  a  rule  that 
those  ending  in  5  or  0  are  overstated,  while  the  adjacent 
ones  are  understated.  But  if  the  age  years  are  united  in 
groups  of  five,  for  instance,  so  that  the  years  with  round 
numbers  come  in  the  middle  of  the  group,  the  errors  coun- 
terbalance each  other."** 

^'  Just  as  quantitative  observation  data  are  arranged  in  magni- 
tude classes,  so  qualitative  data  are  classified  into  groups  with 
peculiar  characteristics.  Thus,  in  statistics  of  occupation  certain 
vocations,  etc.,  constitute  the  classes;  in  trade  statistics  the  various 
wares  are  so  arranged.  But  since  qualitative  data  cannot  furnish 
an  average,  such  groups  need  not  be  considered  further. 

"a  Cf.  "  A  Discussion  of  Age  Statistics,"  by  A.  A.  Young  in  Census 
Bulletin  No.  13,  and  also  "  The  Age  Keturns  of  the  Twelfth  Census," 


84  STATISTICAL  AVERAGES  IN  GENERAL 

To  be  sure,  it  is  possible  to  go  too  far  in  the  formation 
of  groups.  The  characteristic  features  of  the  material  may 
be  lost  in  the  leveling  of  details.  Unfortunately,  the  for- 
mation of  groups  does  not  depend  solely  on  methodological 
considerations.  The  more  the  statistics  are  simplified,  the 
less  the  cost  of  preparation  and  publication.  Thus,  finan- 
cial considerations  are  often  of  influence,  which  is  especially 
the  case  with  official  statistics. 

For  all  these  reasons,  it  will  be  readily  understood  that 
statistical  masses,  especially  such  as  deal  with  single  obser- 
vations, are  normally  expressed  in  magnitude  classes  and 
only  exceptionally  in  full  detail.  The  limited  amount  of 
space  available  also  makes  it  impossible  to  publish  all  the 
individual  data  about  such  matters  as  income,  wages,  etc. 
But  if  a  mass  is  represented  in  magnitude  classes,  only  as 
many  frequencies,  or  numbers  of  items,  need  be  given  as 
there  are  classes,  with  of  course  a  statement  of  the  limits 
of  each  class. 

The  formation  of  magnitude  classes  may  proceed  accord- 
ing to  various  principles.  First,  classes  may  be  formed 
M^hose  limits  are  determined  previously  (without  reference 
to  the  number  of  items  falling  in  each).  In  such  cases, 
the  extent  or  width  of  the  classes  is  also  previously  de- 
termined. Generally  the  same  extent  is  chosen  for  each 
class;  but  classes  may  also  be  formed  of  unequal  extent, 
following  some  natural  formation. ^^  Secondly,  classes  may 
be  formed  to  contain  the  same  number  of  items;  in  these 

by  W.  B.  Bailey  and  J.  H.  Parmelee,  "The  Census  Age  Question," 
by  A.  A.  Young,  and  "The  Census  Age  Question:  A  Reply,"  by 
W.  B.  Bailey  and  J.  H.  Parmelee  in  the  Quarterly  Publications  of 
the  American  Statistical  Association,  Nos.  90,  92,  and  93  (June, 
1910;  December,  1910,  and  March,  1911) .—Translator. 

"  Thus,  in  statistics  of  industry  and  agriculture,  groups  of  similar 
scope  may  not  be  formed,  but  rather,  groups  which  correspond  to 
certain  types  (small,  medium-sized,  and  large  industries,  etc.).  The 
magnitude  classes  of  districts  are  often  similarly  demarcated. 


FORMATION  OF  MAGNITUDE  CLASSES  85 

cases  the  limits  of  the  classes  are  ascertained  through  ex- 
amination of  the  grouping  of  the  items. 

If  magnitude  classes  are  formed  with  previously  estab- 
lished boundaries,  the  statistical  series  indicates  the  fre- 
quency or,  in  other  words,  how  many  items  belong  to  the 
various  classes;  it  will  then  be  generally  found,  both  in 
classes  of  equal  and  in  those  of  unequal  extent,  that  the 
number  of  items  varies  for  the  different  classes.  If,  for 
instance,  wage  classes  are  formed:  (1)  up  to  $1.50,  (2)  $1.50 
to  $2,  (3)  $2  to  $2.50,  etc.,  the  series  will  indicate  the 
number  of  laborers  in  each  class,  and  as  a  rule  these 
numbers  will  differ  from  each  other. 

On  the  basis  of  a  series  which  consists  of  magnitude 
classes  with  previously  fixed  boundaries,  we  may  proceed 
to  form  "  cumulative  "  magnitude  classes,  such  as  are  fre- 
quently employed  by  English  and  American  statisticians 
in  dealing  with  wages.  For  this  purpose  the  numbers  of 
the  items  belonging  to  the  several  magnitude  classes, 
starting  from  the  lowest  or  the  highest,  are  successively 
added,  and  the  sums  thus  obtained  are  indicated.  For 
example,  instead  of  indicating  how  many  laborers  belong 
to  each  of  the  wage  classes  with  fixed  limits,  we  may, 
starting  from  the  lowest,  indicate  how  many  laborers  earn 
up  to  $1.50,  how  many  up  to  $2,  how  many  up  to  $2.50, 
etc. ;  or  starting  from  the  highest  we  may  indicate  how 
many  earn  $2.51  or  more,  $2.01  or  more,  $1.51  or  more, 
etc.  This  latter  method  is  the  commoner.  The  sums  sig- 
nify in  this  case  the  numbers  earning  at  or  above  a  certain 
wage.'""'^^^     Age  data  are  often  cumulated  in  the  same  way 

'"  This  kind  of  group  formation  was  most  freely  employed  in  the 
Special  Report  of  the  Twelfth  (U.  S.)  Census,  entitled  "Employees 
and  Wages,"  prepared  by  Prof.  Davis  R.  Dewey.  Weekly  wages  are 
there  presented  in  50-cent  groups,  and  the  number  of  laborers,  both  in 
absolute  figures  and  in  percentages,  is  given  for  each  class.  The  per- 
centages are  then  worked  over  further  according  to  the  method  in  ques- 
tion, that  is,  proceeding  from  the  highest  wage  class  they  are  summed 
up    successively.     The   resulting   figures   indicate   what   percent   of 


86  STATISTICAL  AVERAGES  IN  GENERAL 

by  indicating  how  many  persons  have  passed  the  various  age 
limits  (so  many  persons  at  and  above  0,  1,  2,  3,  etc.,  years). 

laborers  receive  certain  wages  or  more.  The  column  of  these  figures 
is  called  the  "  Cumulative  Percentage  Column." 

This  kind  of  group  formation  is  especially  valuable  in  comparing 
the  wage  conditions  of  different  years.  The  "  cumulative "  per- 
centages show  clearly  the  evolution  of  wage  conditions,  while  the 
absolute  numbers  and  the  percentages  for  the  individual  wage  classes 
show  apparently  irregular  shiftings  from  which  no  valid  inference 
may  be  drawn.  In  the  above-mentioned  wage  statistics  of  the 
American  Census  Bureau  the  wage  conditions  of  the  years  1900  and 
1890  are  represented  side  by  side  (p.  xxvi).  A  few  examples  may 
be  quoted.  To  the  wage  classes,  $8-8.49,  $8.50-8.99,  $9-9.49,  $9.50- 
9.99,  $10-10.49,  there  belonged  in  the  years  1900  and  1890,  re- 
spectively, the  following  percentages  of  laborers:  0.6  and  0.9,  0.1 
and  0.3,  12.2  and  7.3,  2.9  and  1.1,  3.2  and  5.2.  These  figures  are 
indecisive.  The  cumulative  percentage  column  contains  the  follow- 
ing data  for  the  corresponding  "cumulative"  classes:  in  1900,  72. Ij? 
of  laborers  earned  $8  or  more;  in  1890,  78.1^  (the  percentage  of 
laborers  who  earned  less  than  $8,  therefore,  increased  from  1890  to 
1900  by  6);  in  1900,  71.5jJ  of  laborers  earned  $8.50  or  more;  in 
1890,  77.2^;  in  1900,  71.4^  of  laborers  earned  $9  or  more;  in 
1890,  76.9^,  etc.  These  cumulative  percent  figures  leave  no  doubt 
that  wages  in  1890  were  higher  than  in  1900. 

'"a  There  has  been  some  difference  of  opinion  in  regard  to  the  rela- 
tive level  of  wages  in  the  United  States  in  1900  as  compared  with 
1890.  Prof.  H.  L.  Moore  summarized  the  wage  data  of  the  census  re- 
port on  Employees  and  Wages  in  30  selected  industries,  and  found 
that  the  relative  wages  had  declined  from  100.0  to  99.6  in  the 
decade.  (See  "The  Variability  of  Wages"  in  the  Pol.  Sci.  Quar., 
March,  1907.)  The  index  number  of  the  Bureau  of  Labor  for  rates 
of  pay  per  hour  shows  an  increase  from  100.3  to  105.5  for  the 
same  period.  Prof.  W.  C.  Mitchell  has  examined  the  figures  and 
methods  used  in  the  two  computations  (see  "The  Trustworthiness 
of  the  Bureau  of  Labor's  Index  Number  of  Wages  "  in  the  Quar. 
Jour,  of  Econs.  for  May,  1911),  and  has  concluded  that  because 
the  Bureau  of  Labor  covered  industries  not  included  by  Moore,  and 
because  the  former  took  account  of  the  reduction  of  working  time 
by  using  hourly  wages,  "  the  results  which  Professor  Moore  deduced 
from  Professor  Dewey's  report  afford  no  reason  for  doubting  that 
the  Bureau  of  Labor's  index  number  represents  fairly  the  trend 
of  wages  in  manufactuing  industries." — Translator. 


FORMATION  OF  MAGNITUDE  CLASSES  87 

Such  series  makes  comparisons  easy ;  ^°  they  have  been  much 
employed  by  Galton  and  other  English  statisticians  for 
graphic  representations.  They  have  for  this  purpose  the 
special  advantage  of  producing  a  constantly  rising  curve, 
whereas  the  usual  frequency  curves  rise  and  fall.  The 
graphic  representation  of  such  series  may  also — as  will  be 
shown  later — serve  to  determine  the  median  and  the  mode. 
If  magnitude  classes  with  previously  fixed  limits  are  to 
be  formed,  the  average  of  the  series  may  be  used  as  a  start- 
ing point,  and  the  parts  above  and  below  the  average 
formed  into  groups,  in  such  a  way  that  the  average  is 
the  boundary  between  two  of  the  groups.  But  this  method 
is  not  often  adopted ;  it  is,  however,  frequently  employed 
in  the  division  of  relative  numbers  which  are  to  be  repre- 
sented in  chart  form;  in  such  cases  the  parts  above  and 
below  the  average  are  generally  colored  differently  and  the 
varying  intensity  is  indicated  by  shading.*^  If  different 
phenomena  are  to  be  represented  simultaneously  in  this 
way,  the  kind  of  group  formation  must,  as  a  rule,  be  de- 
termined separately  for  each  chart.  The  same  color  dis- 
tinctions will  then  not  signify  the  same  degrees  of  intensity 
in  the  different  charts.  The  same  colors  may  indicate  in 
one  case  great,  in  another  case  slight,  deviations  from  the 
mean.  In  this  way  the  reader,  who  looks  at  the  different 
supplementary  charts,  easily  gets  false  impressions.  Cheys- 
son  in  his  Album  de  Statistique  agricole  tried  to  remedy 
this  difficulty  by  dividing  into  groups,  not  the  single  rela- 
tive numbers,  but  their  distances  from  the  mean,  and  to  do 
this  in  the  same  way  for  a  whole  series  of  charts.  Thus, 
similar  colors  meant  equal  distances  from  the  mean.     Ber- 

""  Cf.  the  note  79a. 

••  G.  V.  Mayr  says  ( Theoretische  Statistik,  p.  112)  :  "This  manner 
of  presentation  must  be  pronounced  wrong  in  principle.  In  resolving 
the  total  results  of  a  district  into  geographical  details  there  is  no 
reason  why  such  a  decisive  influence  should  be  accorded  to  the  magni- 
tude of  the  general  average." 


'88  STATISTICAL  AVERAGES  IN  GENERAL 

tillon  remarks  in  this  connection:  "  This  process  is  ingeni- 
ous and  very  logical,  but  besides  being  laborious,  it  does 
not  seem  to  have  produced  as  good  results  as  might  have 
been  expected,  because  it  limits  the  means  of  expression, 
already  very  poor,  which  the  shadings  possess."  ^^ 

From  series  of  single  observations  there  arises,  as  has 
been  shown,  by  the  formation  of  magnitude  classes  with 
previously  fixed  limits,  a  series  which  gives  the  number  of 
items  belonging  to  the  individual  classes.  But  often, 
simultaneously  with  this  series,  a  second  series  is  obtained 
by  adding  the  magnitudes  of  the  observation  element,  which 
are  inserted  in  the  several  classes.  Thus,  if  establishments 
are  divided  into  magnitude  classes  according  to  the  number 
of  laborers,  we  obtain,  first,  a  series  which  tells  the  number 
of  establishments  belonging  to  each  class;  but,  secondly, 
we  may  also  compute  how  many  laborers  in  all  are  em- 
ployed in  the  establishments  of  the  different  classes.  If 
we  divide  farms  according  to  their  area,  we  learn  how 
many  farms  belong  to  the  various  classes,  and  at  the  same 
time  we  may  find  out  the  total  area  in  each  magnitude 
class.  In  the  same  way,  income  statistics  regularly  give, 
not  only  the  number  of  people  investigated,  but  also  the 
sum  total  of  the  incomes  in  the  different  classes. 

But,  as  we  have  noted,  other  magnitude  classes  than  those 
of  previously  determined  extent  may  be  formed  from  series 
of  quantitative  single  observations.  Instead  of  fixing  the 
limits  of  the  classes  previously  in  order  to  ascertain  the 
number  of  items  in  each  class,  the  whole  mass  of  items  may 
be  divided  into  equal  parts  and  the  limits  may  be  thus 
determined.  Each  of  these  parts  contains  the  same  number 
of  items,  but  the  range  in  which  these  items  lie,  may  vary. 
If  a  series  is  divided  into  four  equal  groups,  the  values 
forming  the  boundaries  between  the  different  groups  are 
called  quartiles ;  the  quartile  between  the  second  and  third 
group  is  at  the  same  time  the  median  of  the  series.  If 
'"  Cours  6l6nientaire  de  Statistique,  p.  142. 


FORMATION  OF  MAGNITUDE  CLASSES  89 

the  series  is  divided  into  ten  groups,  the  values  which  form 
the  boundaries  are  called  deciles;  if  there  are  a  hundred 
groups,  percentiles.  In  the  last  case  the  method  of  division 
is  often  called  the  "  method  of  percentiles,"  or  "  Galton's 
method,"  after  Galton,  who  has  frequently  employed  it  in 
anthropometry.®^ 

In  order  to  form  equal  parts,  the  items  must  be  of  equal 
weight;  but,  normally,  relative  numbers  and  averages 
possess  varying  weights,  for  which  reason  the  method 
of  percentiles  is  not  applicable  to  series  of  the  third  group. 
It  would  be  theoretically  permissible  with  series  of  the 
second  group,  but  would  possess  no  significance  worth  men- 
tioning, since  such  series,  as  a  rule,  do  not  consist  of  a 
sufficiently  large  number  of  members  to  make  the  applica- 
tion of  the  method  profitable.  Moreover,  the  series  of  the 
second  and  third  groups,  in  order  to  be  divided  into  per- 
centiles, would  have  first  to  be  arranged  according  to  the 
numerical  value  of  the  items,  whereas  the  natural  order 
of  these  series  is  generally  different.  It  follows,  then,  that 
the  application  of  the  method  of  percentiles  is  theoretically 
unobjectionable  but  at  the  same  time  of  practical  value 
only  in  series  of  quantitative  single  observations. 

In  order  to  form  equal  groups,  it  is,  theoretically  at  least, 
necessary  to  know  the  whole  series  in  detail.  If  the  series 
already  consists  of  magnitude  classes  with  previously  fixed 
limits  and  hence  of  unequal  size,  it  is  generally  difficult  to 
fix  the  position  of  the  desired  boundaries  (quartiles,  deciles, 

*' Cf.  Galton,  Natural  Inheritance,  p.  46,  and  his  "Application  of 
the  Method  of  Percentiles  to  Mr.  Yule's  Data  on  the  Distribution  of 
Pauperism"  (Journ.  of  the  Roy.  Stat.  Soc,  1896,  p.  392),  also  "As- 
signing Marks  for  Bodily  Efficiency"  (Report  British  Association, 
1899,  p.  475);  Geissler,  "  Uber  die  Vorteile  der  Berechnung  nach 
perzentilen  Graden  "  (Allg.  statist.  Archiv,  II,  2)  ;  Prof.  John  Dewey, 
"Galton's  Statistical  Methods"  (Quarterly  Publications  of  the 
American  Statistical  Association,  New  Series,  No.  8,  1888)  ;  L. 
Gulick,  "  The  Value  of  Percentile  Grades  "  (Quarterly  Publications  of 
the  American  Statistical  Association,  New  Series,  No.  21,  1893). 


90  STATISTICAL  AVERAGES  IN  GENERAL 

percentiles,  etc.).  But  if  we  liave  the  original  statistical 
material,  it  is  just  as  easy  to  divide  it  into  groups  of  equal 
size  as  into  groups  with  predetermined  limits.  To  be  sure, 
the  formation  of  groups  with  predetermined  limits,  and 
especially  the  formation  of  groups  of  equidistant  limits,  will 
in  the  majority  of  cases  be  more  significant,  and  accord- 
ingly this  method  is  much  oftener  applied  than  the  method 
of  percentiles. 

Magnitude  classes  formed  according  to  the  method  of 
percentiles  have  no  relation  to  an  average  except  in  one 
case.  This  case  occurs  where  an  even  number  of  classes 
of  the  same  size  is  formed,  since  the  boundary  between  the 
two  middle  classes  will  then  be  identical  with  the  median. 

Since  series  of  magnitude  classes  are  very  common,  and 
especially  since  quantitative  individual  observations  are 
very  rarely  published  in  full  detail  but  generally  only  in 
magnitude  classes  of  some  kind,  the  statistician  often  has 
the  task  of  determining  the  average  of  a  series  of  such 
classes.  Cases  are  rare  in  which  the  desired  average  of 
such  a  series  is  the  boundary  of  a  magnitude  class  and 
may  thus  be  taken  directly  from  the  series.  Generally  the 
series  does  not  express  the  average,  which  therefore  has  to 
be  computed.  Even  if  the  original  material  were  accessible 
in  all  its  detail,  the  computation  of  an  average  from  it 
would  frequently  be  so  laborious  that  the  statistician  pre- 
fers to  deal  with  the  magnitude  classes.  But  the  computa- 
tion of  an  average  from  magnitude  classes  is  open  to  one 
grave  objection.  The  computation  of  an  average  presup- 
poses, theoretically,  the  knowledge  of  the  actual  individual 
values  of  the  series  or,  at  least,  of  the  values  of  certain 
portions  of  the  series.  In  computing  the  arithmetic  and 
geometric  means,  all  the  items  of  the  series  must  be  utilized. 
The  former  is  obtained  by  adding  all  the  items  and  dividing 
the  sum  by  the  number  of  items  (n).  The  geometrical 
mean  is  found  by  multiplying  all  n  items,  and  taking  the 
nth  root  of  the  product.     A  knowledge  of  all  the  values 


FORMATION  OF  MAGNITUDE  CLASSES  91 

of  the  series  is,  indeed,  not  necessary  in  order  to  obtain  the 
median  or  the  mode.  Generally  we  can  easily  ascertain 
in  what  magnitude  class  these  two  values  are  contained, 
but  to  determine  them  more  exactly,  it  is  necessary  to 
consider  the  grouping  of  the  individual  values  in  the  class 
concerned. 

In  order  to  compute  averages  from  series  which  contain 
only  magnitude  classes,  we  must,  accordingly,  have  recourse 
to  auxiliary  methods,  especially  to  hypotheses  about  the  dis- 
tribution of  the  values  in  the  different  classes.  These  auxil- 
iary methods,  which  differ  according  to  the  kind  of  average 
sought,  will  be  taken  up  in  the  discussion  of  the  various 
averages. 


CHAPTER  VI 
NATURE  AND  PURPOSE  OF  AVERAGES 

G.  von  Mayr  describes  the  function  of  averages  as  fol- 
lows: "  The  structure  of  a  social  mass,  which  has  found 
expression  in  a  series,  can  only  be  thoroughly  understood 
by  a  careful  study  of  all  its  members.  But  the  more  numer- 
ous the  members  and  groups  of  members,  the  more  im- 
pelling is  the  desire  for  concentrated  information,  that  is 
to  say,  for  a  single  simple  expression  which  contains  in 
itself  the  net  result  of  the  whole  series.  This  is  the  pur- 
pose of  averages."^*  He  speaks  of  an  average  also  as  a 
"  short  expression  of  the  phenomenon  which  levels  all 
differences  of  the  individual  members  of  the  series. ' '  ^^ 
Bowley  expresses  himself  in  a  similar  way:  "  By  the  use 
of  averages  complex  groups  and  large  numbers  are  pre- 
sented in  a  few  significant  words  or  figures, ' '  ^^  and  ' '  The 
object  of  a  statistical  estimate  of  a  complex  group  is  to 
present  an  outline,  to  enable  the  mind  to  comprehend  with 
a  single  effort  the  significance  of  the  whole.  "^^ 

According  to  these  citations,  the  essential  nature  of  aver- 
ages consists  in  describing  series  of  divergent  individual 
values  by  means  of  a  simple  comprehensive  expression. 
The  question  now  arises,  for  what  concrete,  methodological 
purposes  we  need  such  simple  expressions  and  to  what 
ends  we  can  apply  them,  considering  their  nature.  As  we 
shall  show  presently,  various  purposes  may  be  pursued  in 
the  computation  of  averages. 

»*  Theoretische  Statistik,  p.  98.  «» Ibid.  p.  84. 

^'  Elements  of  Statistics,  2nd  ed.,  p.  107.  ^'  Ibid.  p.  7. 

92 


NATURE  AND  PURPOSE  OF  AVERAGES  93 

An  average  may  be  computed  for  its  own  sake,  merely 
to  obtain  a  comprehensive,  characteristic  expression  for  a 
series  of  divergent  values.  But  it  is  often  found  as  a  means 
to  another  end,  mainly  for  purposes  of  comparison,  or 
in  order  to  judge  the  individual  values,  or  in  order  to 
measure  the  dispersion  of  series. 


A.  THE  COMPUTATION  OF  AVERAGES   FOR  THEIR 

OWN  SAKE 

Every  average  characterizes  a  series  in  a  definite  way 
and  also  gives  a  measure  of  the  complex  of  causes  affect- 
ing the  phenomenon  in  question.  An  average  may  be  com- 
puted with  the  sole  purpose  of  obtaining  that  information 
which  the  average  from  its  nature  is  able  to  transmit. 
This  information  differs,  however,  according  to  the  kind 
of  average.  The  arithmetic  mean,  the  median,  the  mode, 
etc.,  each  give  a  different  kind  of  information  about  the 
scries  from  which  they  are  obtained.  The  special  character- 
istics of  the  different  averages  and  the  kind  of  information 
they  give  about  a  series  will  be  discussed  in  the  scond 
part  of  this  book. 

B.  AVERAGES   FOR  PURPOSES   OF  COMPARISON 

1.    GENERxU.  REASONS  FOR  THE  APPLICATION  OF  AVERAGES  FOR 
PURPOSES  OF  COMPARISON 

To  compare  series  of  individual  values  of  any  kind  is 
often  very  difficult,  especially  if  each  series  consists  of 
numerous  members  and  if  a  considerable  number  of  series 
are  to  be  compared.  The  comparison  is  made  easier  if  the 
series  to  be  compared  are  compressed  into  a  few  magnitude 
classes.  But  frequently  that  is  not  enough.  In  the  place 
of  the  different  series,  their  averages  are  then  taken  (arith- 
metic mean,  or  median,  or  mode,  etc.)  ;  these  may  then  be 


94  STATISTICAL  AVERAGES  IN  GENERAL 

compared,  so  to  speak,  at  a  glance.  Let  us  suppose  that 
a  comparison  is  desired  of  the  age  conditions  of  those  who 
marry  in  different  countries  or  in  different  social  groups. 
Even  if  the  age  tabulation  of  the  masses  to  be  compared 
is  available  in  absolute  numbers,  a  comparison  of  these 
series  is  naturally  out  of  the  question.  And  even  if  series 
of  subordinate  numbers  occur,  the  comparison  will  hardly 
be  possible.  Accordingly,  the  attempt  will  be  made  first 
to  compress  the  series  into  a  few  magnitude  classes.  If  a 
survey  is  still  impossible,  the  average  age  at  marriage  for 
the  different  lands  or  groups  of  population  will  be  com- 
puted and  these  values  compared  with  each  other. 

Averages  are  very  often  applied  when  series  of  quanti- 
tative individual  observations,  which  refer  to  different 
times,  countries,  or  groups  of  population,  are  to  be  com- 
pared; instead  of  a  detailed  tabulation  of  the  ages  of 
people  in  different  times,  or  belonging  to  different  groups 
of  population,  the  average  ages  are  very  simply  compared ; 
the  same  process  is  followed  if  wages,  incomes,  heights,  etc., 
are  substituted  for  ages.  It  is  extremely  difficult,  for 
instance,  to  compare  several  mortality  tables,  but  it  is 
easy  to  compare  the  average,  probable  or  normal  lengths 
of  life  computed  from  those  tables. 

In  comparing  averages  it  should  always  be  borne  in 
mind  that  the  comparison  extends  only  to  those  qualities 
of  the  series  which  the  average  is  capable  of  reproducing, 
but  that  the  other  differences  between  the  series  cannot  be 
determined  by  the  comparison  of  these  averages.  If,  for 
example,  we  compare  the  modes  (that  is,  the  relatively 
most  frequent  values)  of  two  series  of  wage  data,  then  this 
comparison  is  of  course  restricted  to  those  qualities  of  the 
two  series  which  can  be  expressed  by  the  mode.  Therefore, 
that  series  which  possesses  the  higher  mode  may  at  the 
same  time  have  the  lower  arithmetic  mean,  and  vice  versa. 
Also  series  which  agree  in  regard  to  an  average  may  at 


NATURE  AND  PURPOSE  OF  AVERAGES  95 

the  same  time  have  a  different  dispersion  or  grouping  of 
the  items  about  the  average. 

Often,  averages  are  computed  for  series  which  refer  to 
long,  successive  periods  of  time,  and  compared  in  order  to 
determine  whether  the  phenomenon  in  question  is  subject 
to  a  distinct  evolutionary  tendency.  This  question  can 
frequently  not  be  answered  from  the  irregular  fluctuations 
of  individual  years.  But  if  averages  are  compared  for 
longer  periods,  the  unessential  fluctuations  of  the  individual 
years  are  eliminated  and  the  essential  changes  revealed. 

The  comparison  of  demographic  numbers  is  usually  equiv- 
alent to  a  comparison  of  averages.  Relative  numbers  are 
commonly  computed  exclusively  for  purposes  of  compari- 
son, since  absolute  numbers  are  not  generally  suited  for 
this.  If  we  wish,  for  instance,  to  compare  the  birth  rate 
of  two  countries,  the  comparison  of  the  absolute  numbers 
of  births  in  the  two  countries  would  lead  to  no  conclusion. 
Countries  of  different  sizes,  of  course,  do  not  have  the 
same  numbers  of  births.  Only  by  dividing  the  births  in 
each  case  by  the  population,  and  thus  eliminating  the  in- 
fluence of  populations  of  different  sizes,  do  we  obtain  com- 
parable values. 

The  comparison  of  averages  generally  consists  in  merely 
placing  them  side  by  side;  in  large  numbers  it  is  useful 
to  compute  the  difference  between  the  two  figures  in  order 
to  spare  the  reader  this  task.  Often  the  two  figures  are 
brought  into  a  percent  relation.  Thus,  the  average  daily 
sick  rate  is  usually  given  as  a  percentage  of  the  average 
number  liable  to  sickness.  Finally,  the  difference  between 
the  two  figures  compared  may  be  given  as  a  percentage 
of  the  larger  or  smaller  one. 

2.    MEAN    INDEX   NUMBERS 

A  unique  example  of  the  application  of  averages  for 
purposes  of  comparison  in  time  relations  is  furnished  by 


96  STATISTICAL  AVERAGES  IN  GENERAL 

mean  index  numbers.  These  are  computed  on  the  basis  of 
statistically  determined  changes  of  parts  of  a  mass,  in  order 
to  find  out  the  change,  not  directly  measurable,  which 
affects  the  mass  as  a  whole. 

The  best  known  application  of  this  method  is  the  com- 
putation of  mean  index  numbers  of  prices  in  order  to  com- 
pare the  level  for  different  years.  The  prices  of  individual 
commodities  are  subject  to  special  influences,  depending 
on  conditions  of  production  and  sale  and,  therefore,  fre- 
quently show  quite  independent  fluctuations,  or,  it  may  be, 
even  opposite  tendencies.  But  it  may  be  significant  to 
determine  the  net  result  of  all  these  numerous  individual 
movements  and  thus  to  ascertain  the  tendency  of  the  general 
level  of  prices.  For  this  purpose,  the  movements  of  prices 
of  the  individual  commodities  are  given  relatively  to  the 
prices  of  a  standard  year  or  to  the  average  prices  of  a 
standard  period  by  means  of  ratios  or  index  numbers,  and 
from  these  an  average  called  the  mean  index  number  is 
computed  for  each  year.^^ 

A  very  extensive  literature  has  grown  up  concerning  in- 
dex numbers  and  the  problems  arising  from  them.  Both 
the  British  Association  for  the  Advancement  of  Science 
and  the  International  Statistical  Institute  have  gone  into 
the  question  thoroughly.  The  question  at  issue  is  what 
average  (simple  or  weighted  arithmetic  mean,  geometric 
mean,  median,  etc.)  should  be  used  in  combining  the 
index  numbers  for  the  individual  commodities.  If  the 
weighted  arithmetic  mean  is  chosen,  the  question  arises, 
How  shall  we  determine  the  "  weights  "  to  be  applied?  We 
will  return  to  these  questions  in  the  second  part  of  the  book 

*'  Some  authors  limit  themselves  to  adding  the  individual  indices; 
for  instance,  The  Economist  and  Krai.  An  opponent  of  mean  index 
numbers  in  general  is  the  Dutch  economist,  N.  G.  Pierson  (see  his 
"  Further  Considerations  on  Index  Numbers  "  in  The  Economic  Jour- 
nal, 1896,  p.  127  ff.;  cf.  also  the  reply  of  Edgeworth,  "A  Defence 
of  Index  Numbers,"  ibid.  p.  132  ff.). 


NATURE  AND  PURPOSE  OF  AVERAGES     97 

when  we  consider  the  various  kinds  of  averages.  Further- 
more, we  must  determine  what  commodities  we  are  going 
to  select.  The  Economist,  for  instance,  considers  only  22, 
Sauerbeck  35  (with  45  index  numbers  because  of  the  repeti- 
tion of  certain  important  ones),  Sotbeer  114,  Falkner  223, 
and  the  United  States  Bureau  of  Labor  258.  For  these 
commodities,  in  turn,  various  prices  may  be  used  (average 
prices  or  single  quotations,  wholesale  or  retail  prices),  which 
again  may  depend  on  various  sources,  such  as  trade  journals, 
statements  of  merchants,  etc.  The  standard  year  or  the 
standard  period  has  also  to  be  chosen.  The  selection  of 
commodities,  of  the  method  of  averaging,  of  the  source 
of  prices,  and  of  the  standard  period  must  be  largely  de- 
termined by  the  conditions  surrounding  particular  in- 
vestigations. However,  we  cannot  investigate  all  of  these 
conditions  here.^''"^^'' 

Llean  index  numbers  may  be  utilized  to  determine  other 
general  tendencies  besides  the  movement  of  prices.     Thus, 

"  Cf.  the  article  "  Preis  "  and  (especial]}')  the  chapter  "  Statis- 
tische  Bestimmung  des  Preisniveaus "  by  R.  Zuckerkandl  in  the 
Handw.  d.  Staatsw.  (with  bibliography);  also  the  article  "Index 
Numbers "  by  Edgeworth  in  Palgrave's  Dictionary  of  Political 
Economy  and  the  article  "  Price-Levels  "  in  Mulhall's  Dictionary  of 
Statistics;  see  also  Tabellen  zur  Wahrungsstatistik  of  the  Min- 
istry of  Finance  (Vienna),  2nd  ed.,  Pt.  II,  4th  number  (Prices, 
Wages,  Purchasing  Power  of  Money),  p.  919  ff.  The  special  sta- 
tistical questions  are  treated  particularly  by  Bowley  in  Elements 
of  Statistics,  Chap.  IX.  Cf.  also  "  Bericht  iiber  die  Tatigkeit  des 
statistischen  Seminares  der  Universitat  Wien  im  Wintersemester, 
1903-1904,"  review  by  J.  Schumpeter  of  the  method  of  index  numbers 
(Stat.  Monatsschrift,  1905,  p.  191  ff.)  and  Report  on  Wholesale 
and  Retail  Prices  (London,  1903),  Appendix  II. 

'°a  A  recent  summary  of  the  literature  on  index  numbers  is  given-in 
the  Canadian  Department  of  Labor  report  on  Wholesale  Prices  in 
Canada,  1890-1909.  Prof.  Irving  Fisher  has  given  a  thorough  dis- 
cussion of  "  The  Best  Index  Numbers  of  Purchasing  Power "  in 
Chap.  X  and  Appendix  of  "  The  Purchasing  Power  of  Money," 
Macmillan,  1911. — Tbanslatob. 


98  STATISTICAL  AVERAGES  IN  GENERAL 

Bowley  has  used  them  freely  to  determine  the  changes  in 
the  level  of  wages  for  large  districts  on  the  basis  of  known 
changes  in  the  wages  of  certain  localities,  and  also  to  de- 
termine changes  in  the  wage  level  of  certain  industries 
from  what  is  known  as  to  the  movement  of  wages  of  certain 
occupations  within  those  industries.  For  instance,  he  has 
presented  the  evolution  of  the  wages  of  agricultural  la- 
borers of  diiferent  parts  of  Great  Britain  and  Ireland  by 
indices,  and  taking  the  average  of  the  latter  he  has  com- 
puted the  index  number  for  the  whole  United  King- 
dom,*^" 

In  the  same  way  Bowley  has  also  used  indices  to  show 
the  tendency  of  the  wages  of  book  printers  in  various 
cities  and  then  by  computing  the  weighted  arithmetic 
mean  has  obtained  the  index  for  the  whole  United  King- 
dom.^^  Bowley  has  also  computed  index  numbers  of  wages 
in  26  occupations  connected  with  the  building  of  machines 
and  ships  for  18  centers  of  industry.  He  has  then  com- 
bined these  index  numbers  into  mean  index  numbers,  first, 
according  to  occupation  for  the  whole  kingdom,  second, 
without  distinction  of  occupation  for  the  18  centers  of 
industry.  Finally,  from  these  mean  index  numbers  he 
obtained  grand  mean  indices  by  computing  the  simple  and 
weighted  arithmetic  means.  The  grand  mean  indices  show 
the  tendency  of  wages  for  the  laborers  of  the  entire  machine 
and  ship  building  industry  of  the  whole  kingdom.^^ 
Bowley  has,  furthermore,  obtained  comprehensive  mean  in- 
dex numbers  for  the  movement  of  wages  in  the  British 
building   industry.^*     George   H.   Wood,   his   collaborator, 

'"  The  Statistics  of  Wages  in  the  United  Kingdom  during  the  Last 
Hundred  Years,  Pt.  IV,  "Agricultural  Wages"  (Journ.  of  the  Koy. 
Stat.  Soc,  Vol.  LXII,  1899,  especially  p.  568  ff.). 

»^  Ibid.  pp.  708-715. 

»=  Ibid.  Vol.  LXIX,  1906,  p.  158  ff. 

»=  Ibid.  Vol.  LXIV,  1901,  p.  106  ff. 


NATURE  AND  PURPOSE  OF  AVERAGES  99 

has  also  computed  averages  from  indices  of  wages  of  mis- 
cellaneous occupations.^*"'**^ 

Wood  has  also  undertaken  to  express  the  development  of 
the  consumption  of  the  English  population  by  means  of 
general  index  numbers.^^  The  consumption  of  the  popula- 
tion and  any  change  in  it  can,  of  course,  only  be  obtained 
from  observations  regarding  the  consumption  of  single 
articles.  Wood  has,  accordingly,  represented  the  consump- 
tion per  head  of  the  population  of  England,  1860-1896,  of 
the  more  important  articles  (flour,  meat,  rice,  coffee,  sugar, 
tea,  tobacco,  etc.)  by  means  of  index  figures,  fixing  the 
average  consumption  of  his  standard  period,  1870-1879,  at 

•*"  Changes  in  Average  Wages  in  New  South  Wales,  1823-1898," 
Journ.  of  the  Roy.  Stat.  Soc,  Vol.  LXIV,  1901,  p.  327  ff.  (based  on 
the  data  in  T.  A.  Coghlan's  Wealth  and  Progress  in  New  South 
Wales).  Prof.  Falkner  observes  ("Die  Lohnstatistik  in  der  Theorie 
und  in  der  Praxis,"  Allg.  Stat.  Archiv,  Vol.  VI,  1st  half -vol.,  1902) 
that  he  applied  the  method  of  index  numbers  to  wages  even  before 
Bowley  in  the  Aldrich  Report  of  1893,  and  that  Sir  Robert  Griflfen 
recommended  special  indices  for  wages  in  his  review  of  index  num- 
bers made  for  the  International  Statistical  Institute,  1887.  Isolated 
index  numbers  for  wages  are  also  to  be  found  in  Salaires  ct  Duree 
de  travail  dans  I'lndustrie  frangaise  of  the  year  1897  (for  example, 
Vol.  IV,  p.  277).  Recently  total  index  numbers  have  been  em- 
ployed in  the  wage  statistics  published  annually  by  the  American 
Bureau  of  Labor  in  such  a  way  as  to  embrace  the  indices  for  the 
movement  of  wages  in  the  various  occupations  by  industries  and 
finally  for  the  total  industry.  (See  Bulletin  of  the  Bureau  of  Labor, 
Washington,  July,  1907,  p.  22  f. ) .  Cf.  also  the  suggestions  of  W.  C. 
Mitchell  regarding  the  application  of  index  numbers  in  wage  sta- 
tistics in  Publications  of  the  Amer.  Stat.  Assoc,  Vol.  IX,  p.  325  ff. 

'■•a  The  English  Board  of  Trade  has  recently  used  the  method  of 
index  numbers  for  comparison  of  wages  and  hours  of  labor,  rents  and 
housing  conditions,  retail  prices  of  food  and  the  expenditure  of 
working-class  families  on  food  for  the  United  Kingdom  (see  Report 
Cd.  3864),  with  similar  items  for  Germany  (Cd.  4032),  France 
(Cd.  4512),  Belgium  (Cd.  5065),  and  the  United  States  (Cd. 
5609).— Tbanslator. 

"  "  Some  Statistics  of  Working  Class  Progress  since  ISGO,"  Journ. 
of  the  Roy.  Stat.  Soc,  Vol.  LXII,  1899,  p.  639  S.,  especially  p.  654  f. 


100  STATISTICAL  AVERAGES  IN  GENERAL 

100.  From  these  index  numbers  Wood  computed  the  arith- 
metic mean  and  five  different  weighted  means,  in  which 
the  changes  in  the  total  consumption  of  the  population  are 
expressed. 

Similarly,  Wood  has  also  measured  the  changes  in  the 
amount  of  aid  given  to  the  unemployed  of  the  various 
English  trade  unions  for  the  years  1860-1896  by  means  of 
index  numbers  (taking  as  his  standard  period  1882-1891), 
and  has  combined  these  index  numbers  into  an  average."'' 

Neumann-Spallart  has  attempted  to  solve  a  much  more 
comprehensive  problem  by  the  use  of  mean  index  num- 
bers, namely,  that  of  finding  a  "  measure  of  the  varia- 
tions in  the  economic  and  social  condition  of  nations. ' '  He 
distinguished  for  this  purpose  four  groups  of  symptoms,  the 
first  two  being  economic,  as  affecting  production  and  trade, 
the  third,  socio-economic,  affecting  consumption,  emigra- 
tion, banks,  etc.,  the  fourth,  moral,  affecting  birth  rate, 
percentage  of  illegitimate  births,  suicides,  crimes,  etc.  His 
plan,  as  presented  at  the  meeting  of  the  International 
Statistical  Institute  in  Rome  (1887),  was  to  compute  index 
numbers  for  the  different  symptoms  and  then  to  combine 
these  into  averages,  first,  for  the  several  groups  of  symp- 
toms, second,  for  the  single  countries  and,  finally,  for  the 
six  countries,  England,  France,  Germany,  Austria,  Bel- 
gium, and  the  United  States,  taken  together.^^ 

This  plan,  whose  accomplishment  was  unfortunately  pre- 
vented by  Neumann-Spallart 's  death,  arouses  grave  mis- 
givings. The  indices  from  which  he  wished  to  compute 
averages  do  not  refer  to  complementary  parts  of  a  homo- 
geneous totality  (as,  for  instance,  the  prices  of  the  various 
commodities  which  make  up  a  general  level  of  prices),  but 
to  very  different  social  phenomena;  each  of  these  phenom- 
ena may  be  symptomatic  of  certain  economic  and  social 

"•  "  Trade  Union  Expenditure-  on  Unemployed  Benefits  since  1860," 
Roy.  Stat.  Soc.  Journ.,  Vol.  LXIII,  1900,  p.  88  ff. 
"  Cf.  Bulletin  de  I'Inst.  intern,  de  Stat.,  Vol.  II,  Pt.  I,  pp.  150-159. 


NATURE  AND  PURPOSE  OF  AVERAGES    101 

conditions,  but  it  is  very  questionable  whether,  by  taking 
all  these  phenomena  together,  we  obtain  a  valid  measure 
for  the  general  social  condition  and  its  changes.®^* 

3,  MEANING  OF  THE  POSTULATE  OF  THE  GREATEST  POSSIBLE 
HOMOGENEITY  FOR  THE  COMPARISON  OF  AVERAGES  AND 
RELATIVE  NUMBERS 

The  postulate,  previously  stated,^^  of  the  greatest  possi- 
ble homogeneity  of  series  or  masses  lying  at  the  basis  of 
averages  and  relative  numbers,  has  special  significance  in 
the  comparison  of  averages  and  of  such  relative  numbers 
as  are,  in  reality,  averages.  The  comparison  of  only  those 
averages  and  relative  numbers  which  refer  to  homogeneous 
series  or  masses  yields  reliable  conclusions ;  the  comparison 
of  other  averages  and  relative  numbers  may  easily  mislead 
and  is  only  reliable  under  special  conditions.  This  will 
be  demonstrated  in  the  following  pages,  for  the  most  part 
with  the  same  examples  as  have  already  been  employed  ^* 
to  establish  this  postulate  in  general  in  connection  with  the 
nature  of  averages  and  relative  numbers. 

''a  Roger  W.  Babson  has  recently  combined  a  miscellaneous  lot  of 
statistics  into  a  single  series  of  index  numbers  which  he  calls 
"  Summary  Barometer  Figures."  He  combines  twenty-five  series  of 
statistics,  which  are  grouped  under  the  twelve  headings:  building  and 
real  estate,  bank  clearings,  business  failures,  labor  conditions,  money 
conditions,  foreign  trade,  gold  movements,  commodity  prices,  invest- 
ment market,  crops  and  commodity  statistics,  railroad  earnings,  and 
social  conditions.  Cf.  Business  Barometers  for  Forecasting  Condi- 
tions, published  by  Roger  W.  Babson,  Wellesley  Hills,  Mass.  Jas. 
H.  Brookmire  of  St.  Louis,  in  the  publication  entitled  The  Brook- 
mire  Economic  Charts,  publishes  similar  figures.  However,  Brookmire 
weights  his  individual  indices  according  to  the  importance  that  they 
are  supposed  to  have  in  indicating  economic  crises.  ( See  "  ^lethods 
of  Business  Forecasting  Based  on  Fundamental  Statistics  "  by  J.  H. 
Brookmire  in  the  Am.  Econ.  Rev.,  March,  1913.) — Traxslatob. 
*'  P.  65  ff  »»  P.  65  ff. 


102  STATISTICAL  AVERAGES  IN  GENERAL 

(a)  Postulate  of  the  greatest  possible  homogeneity  in  the 
comparison  of  averages  obtained  from  series  of 
individual  observations 

Series  of  quantitative  single  observations  which  are  not 
homogeneous  are  sometimes  so  because  they  contain  cases 
which  do  not  show  the  element  of  observation  at  all,  the 
causal  complex  being  inoperative  in  such  cases.  The  size 
of  the  arithmetic  mean  of  such  a  non-homogeneous  series 
depends  on  the  ratio  of  the  items  having  a  positive  to 
those  having  a  zero  measurement.  The  same  thing  is  true, 
mutatis  mutandis,  of  certain  other  averages  as,  for  instance, 
the  median.^""  A  comparison  of  averages  from  such  hetero- 
geneous series  is  evidently  unreliable.  That  one  of  the 
series  in  which  the  element  investigated  has  the  greater 
values  may  yield  the  smaller  average,  simply  because  the 
series  contains  relatively  more  items  with  zero  measure- 
ments (null  items).  Let  us  take  as  an  example  the  aver- 
age marital  fecundity.  If  we  compare  the  average  num- 
ber of  children  for  two  countries  from  all  the  marriages, 
fruitful  and  unfruitful,  it  may  happen  that  the  country 
in  which  the  marriages,  when  not  entirely  unfruitful,  pro- 
duce the  greater  number  of  children,  may  show  the  smaller 
general  marital  fecundity  because  of  the  larger  percentage 
of  unfruitful  marriages. 

The  elimination  of  null  items  is,  of  course,  only  feasible 
where  individual  observations  occur.  For  example,  in  com- 
puting the  average  consumption  of  meat  or  alcohol  for 
the  total  population,  it  is  impossible  to  eliminate  those 
persons  or  classes  who  do  not  take  part  in  the  consumption. 
Great  difficulties  must  evidently  arise  from  this  fact  in 
making  comparisons.  If,  for  instance,  two  countries  or 
cities  are  compared,  it  may  happen  that,  in  consequence 
of  different  percentages  of  those  not  taking  part  in  the 
consumption,  the  persons  actually  concerned  in  the  country 

""See  above,  p.  66. 


NATURE  AND  PURPOSE  OF  AVERAGES    103 

or  city  with  the  smaller  average  may  consume  more  than 
in  the  country  or  city  with  the  larger  average.  The  per- 
centage of  individuals  not  concerned  may  evidently  vary 
in  different  places,  since  it  depends  on  many  demographic 
conditions,  not  only  on  the  sex  and  age  structure  of  the 
population,  but  also  on  various  social  and  psychological 
factors.  A  trustworthy  comparison  of  general  averages  will 
only  be  possible  when  we  may  assume  that  the  ration  of  the 
individuals  concerned  to  those  not  concerned  is,  on  the 
whole,  the  same  in  the  masses  to  be  compared.  This  pre- 
requisite will  hardly  ever  be  fulfilled  in  geographical  com- 
parisons, but,  on  the  other  hand,  fairly  often  in  time  com- 
parisons which  do  not  extend  over  too  long  a  period.  Thus, 
we  may  properly  draw  certain  conclusions  from  the  figures 
for  the  average  consumption  of  alcohol  or  meat  for  a  coun- 
try or  city  for  a  term  of  years. 

Series  of  quantitative  single  observations  may  also  be 
regarded  as  non-homogeneous,  if  among  the  items,  so  far 
as  they  are  positive,  special  parts  may  be  distinguished 
which  are  governed  by  different  and  independent  causes. 
The  size  of  the  average  (both  arithmetic  mean  and  certain 
other  averages)  obtained  from  such  a  series  depends  essen- 
tially on  the  proportion  in  which  the  various  more  homo- 
geneous parts  stand  to  one  another."^  Hence,  the  com- 
parison of  such  averages  may  easily  be  misleading.  Let 
us  consider  the  comparison  of  wages  in  two  districts  by 
means  of  general  average  wages.  Suppose  that  all  laborers 
without  distinction  of  sex  have  been  included,  although 
sex  in  the  districts  in  question  has  an  influence  on  the 
amount  of  wages.  Let  us  assume  that  in  district  A  50;^ 
men  and  50^  women  are  employed,  the  wages  of  the 
men  average  $15,  those  of  the  women  $5,  the  average  for 
all  laborers  being  $10;  in  district  B  the  women  constitute 
only  10^  of  the  total  number  of  laborers,  the  wages  of 
the  men  average  $12,  those  of  the  women  $4,  so  that  both 

*"  See  above,  p.  70. 


104  STATISTICAL  AVERAGES  IN  GENERAL 

men  and  women  have  considerably  smaller  wages  than  in 
district  A.  Nevertheless,  the  general  average  wage  in  B 
is  $11.20  because  of  the  smaller  percentage  of  women,  while 
in  A  it  is  only  $10.  Whoever  compares  merely  the  general 
averages  will  be  tempted  to  assume  that  in  B  the  wage  con- 
ditions are  better,  whereas  exactly  the  opposite  is  the  case. 
As  a  matter  of  fact,  false  conclusions  often  occur  in  cases 
which  correspond  to  the  scheme  just  illustrated.^"^ 

This  scheme  may  also  easily  be  extended  to  the  case  where 
within  the  masses  to  be  compared  there  are  more  than 
two  heterogeneous  parts.  For  example,  within  a  mass  of 
laborers  several  categories  may  occur  with  various  wages. 
Thus,  according  to  Austrian  statistics,  the  mine  laborers 
consist  of  pickmen  and  their  helpers,  other  adult  miners, 
adult  day  laborers,  boys  and  women  laborers;  the  wages 
of  these  categories  vary  greatly,  decreasing  in  the  order 
mentioned.  Now  if  we  compute  average  wages  for  all  the 
miners  for  different  years,  the  comparison  of  these  averages 
might  readily  be  misleading.  Though  the  wages  of  each 
category  had  remained  the  same  from  year  to  year,  yet 
the  average  wages  of  all  the  miners  might  have  fallen  in 
ease  the  more  poorly  paid  categories  had  increased.  The 
wages  of  each  category  might  indeed  have  increased  and 
yet  the  general  average  have  fallen.^''^     A  comparison  of 

^"^  The  average  age  at  marriage  in  Germany  offers  an  example 
which  corresponds  to  this  scheme  but  which  substitutes  a  time  for 
a  space  comparison.  This  average  has  fallen  in  the  last  few  years. 
Industrial  laborers  and  the  agricultural  population  form  a  large 
proportion  of  those  who  marry,  the  former  with  a  very  low  and  the 
latter  with  a  relatively  high  age  at  marriage.  Because  of  the 
industrial  development  of  Germany  the  former  constitute  an  ever 
increasing  quota  of  the  total  population.  The  average  age  of  those 
marrying  must  necessarily  fall  merely  by  reason  of  this  circumstance, 
even  though  neither  the  industrial  laborers  nor  the  agricultural 
population  actually  marry  earlier  than  formerly.  (Cf.  G.  v.  Mayr, 
Bevolkerungsstatistik,  p.  401  f. ) 

^*'*  The  following  would  be  a  similar  case:    The  average  number 


NATURE  AND  PURPOSE  OF  AVERAGES    105 

averages  of  series  in  which  there  are  heterogeneous  parts  is 
only  allowable,  therefore,  on  the  assumption  that  the  hetero- 
geneous constituents  are  present  in  equal  proportion  in  the 
masses  to  be  compared.  This  condition  will  most  often 
be  fulfilled  in  the  case  of  time  comparisons  which  do  not 
extend  over  too  great  a  period. 

From  the  above  examples  it  may  be  seen  how  easily 
erroneous  conclusions  may  be  drawn  from  statistical  com- 
parisons. Even  the  trained  statistician  is  exposed  to  errors, 
when  he  does  not  realize  that  the  averages  which  he  com- 
pares refer  to  non-homogeneous  masses.  If  the  heterogene- 
ous groups  which  compose  a  mass  cannot  be  separated  from 
each  other  (for  instance,  because  the  criterion,  according 
to  which  the  items  were  to  be  distinguished,  was  not  con- 
sidered when  the  observation  was  made),  and  if  it  is  also 
not  certain  that  these  groups  are  represented  in  like  pro- 
portion in  the  masses,  then  even  the  best  statistician  is 
unable  to  arrive  at  a  result.  This  also  explains  how  the 
same  statistical  material,  when  differently  handled  and 
grouped,  seems  frequently  to  prove  exactly  opposite  asser- 
tions. Hence  a  certain  popular  skepticism  about  statistics 
"  which  can  prove  anything."  The  statistical  method 
really  demands  unusual  accuracy  and  conscientiousness. 
Not  seldom  does  the  trained  statistician  recognize  the  in- 
sufficiency of  his  material  and,  refraining  from  positive 
assertions,  confines  himself  to  merelj^  hypothetical  conclu- 
sions, while  the  untrained  layman  draws  false  conclusions 
from  the  statistical  data. 

of  inhabitants  per  house  may  have  increased  from  one  census  to 
another.  May  we  conclude  from  this  that  a  house,  on  the  average, 
now  contains  more  people  and  that  the  population  is  living  closer 
together?  Certainly  not,  for  larger  houses  may  have  been  built  be- 
tween the  two  censuses.  Similarly,  it  would  be  insufficient  to  com- 
pare the  average  number  of  pupils  per  school  at  two  different  times 
without  reference  to  possible  changes  in  the  number  of  classes  of 
which  the  schools  consist. 


106  STATISTICAL  AVERAGES  IN  GENERAL 

(b)  Postulate  of  the  greatest  possible  homogeneity  in  the 

comparison  of  averages  from  series  whose  members 
indicate  the  size  of  masses  limited  in  a  definite  way 

Among  the  series  of  the  second  group,  whose  members 
indicate  the  size  of  definitely  limited  masses,  time  series  are 
the  most  important.  From  time  series,  as  we  have  already 
mentioned,  averages  for  parts  may  be  computed  and  com- 
pared with  each  other  in  order  to  ascertain  whether  the 
series  shows  a  definite  tendency.  It  may,  under  certain 
conditions,  be  possible  to  form  and  compare  parts  of  time 
series  which  are  homogeneous  in  themselves  but  differ  from 
each  other  in  regard  to  their  causation.  In  such  compari- 
sons everything  depends  on  the  kind  of  division.  If  the 
line  of  demarcation  is  wrongly  drawn,  it  may  be  that  no 
conclusion  is  possible,  or  else  a  false  conclusion  may  be 
reached.  If,  on  the  other  hand,  we  succeed  in  comparing 
homogeneous  time  divisions,  valuable  information  may  be 
obtained. 

(c)  Postulate  of  the  greatest  possible  homogeneity  in  the 

comparison  of  relative  numbers 

The  same  difficulties  as  in  the  comparison  of  averages 
from  non-homogeneous  series  recur  in  the  comparison  of 
relative  numbers  which  have  arisen  from  the  subdivision 
or  interrelation  of  non-homogeneous  masses. 

The  comparability  of  relative  numbers  may,  first  of  all, 
be  impaired  by  the  fact  that  in  computing  them  the  parts 
with  zero  measurement  were  not  eliminated."*  The  numer- 
ical value  of  the  relative  numbers  will  then  depend  on 
the  frequently  varying  proportion  of  the  null  items.  For 
instance — to  cite  first  a  subordinate  number — the  propor- 
tion of  the  unmarried  in  a  country  depends  largely  on  the 
number  of  those  who  have  not  yet  reached  the  marriage- 

'»*  See  above,  p.  73. 


NATURE  AND  PURPOSE  OF  AVERAGES  107 

able  age.  If,  therefore,  we  compare  the  percentage  of  the 
unmarried  in  two  countries,  the  higher  percentage  in  one 
country  may  be  due  to  a  relatively  larger  number  of  chil- 
dren, while  among  those  capable  of  marrying  the  unmarried 
may  not  be  more  numerous.  A  trustworthy  conclusion  can- 
not be  drawn  until  the  percentage  of  the  unmarried  is 
determined  and  compared  for  those  of  marriageable  age 
in  the  two  countries. 

For  similar  reasons  the  general  frequency  figures,  which 
arise  from  the  interrelation  of  definite  events  (births,  mar- 
riages, crimes,  etc.)  with  the  total  population,  are  hardly 
comparable.  Their  numerical  value  depends  chiefly  on  the 
ratio  of  the  positive  to  the  null  parts,  which  ratio  may 
vary  from  case  to  case.  In  the  country  A  the  marriageable 
population  may  show  a  higher  marriage  rate  than  in  the 
country  B.  Yet  the  country  A  may  have  the  lower  general 
marriage  rate,  if  the  unmarriageable  age  classes  are  rela- 
tively larger  than  in  B.  Thus,  the  mere  comparison  of  such 
general  frequency  figures  may  be  wholly  misleading,  and 
for  purposes  of  comparison  those  "  specific  "  frequency 
figures  are  regularly  to  be  preferred  which  arise  from  the 
interrelation  of  the  events  with  the  parts  of  the  population 
having  a  positive  measurement.^"**  General  frequency  fig- 
ures are  comparable  only  when  the  percentage  of  the  null 
parts  of  the  population  is  the  same  in  the  cases  compared. 
This  is  hardly  ever  the  case  in  comparing  different  countries, 
and  is  permissible  in  time  comparisons  for  the  same  country 
only  for  limited  periods.^"* 


»o<a  There  are,  however,  certain  purposes  for  which  the  null  items 
need  not  be  eliminated.  The  crude  rates  are  the  very  figures  we 
desire  when,  for  instance,  we  wish  to  compare  results  for  two  cen- 
turies regardless  of  the  various  causes  contributing  to  those  results. 
Then,  too,  in  many  cases  statisticians  are  forced  to  use  crude  rates 
because  the  data  for  the  computation  of  more  refined  rates  are  not 
available. — Translator. 

*'"  Cf.  Westergaard's  discussion  of  cases  in  which  the  general  mar- 


108  STATISTICAL  AVERAGES  IN  GENERAL 

In  computing  relative  numbers  not  only  is  the  elimination 
of  null  parts  desirable  but  also  the  division  of  the  masses 
into  more  homogeneous  parts,  primarily,  as  already  men- 
tioned,^"*^ in  order  to  obtain  relative  numbers  corresponding 
to  a  definite,  unified  causal  complex,  and  also,  as  will  now 
be  shown,  to  make  comparisons  easy.^°^* 

The  numerical  value  of  relative  numbers  based  on  non- 
homogeneous  masses  depends  essentially  on  the  proportion 
in  which  the  various  parts  of  different  subdivisions  or  of 
different  intensity  stand  to  one  another.  If  this  proportion 
differs  in  the  masses  to  be  compared,  then  a  comparison  is 
very  difficult  if  not  quite  impossible.  We  shall  give  two 
examples,  one  for  subordinate  and  one  for  coordinate  num- 
bers. 

Let  us  assume  that  the  comparison  of  two  populations 
(exclusive  of  those  not  yet  of  marriageable  age)  in  regard 
to  conjugal  condition  has  indicated  that  population  A  pos- 
sesses a  considerably  larger  percentage  of  widows  than 
population  B.  The  question  now  arises  whether  a  definite 
conclusion  is  permissible  on  the  basis  of  this  comparison. 
To  ascertain  this,  let  us  try  to  form  more  homogeneous  parts 
in  the  two  masses  to  be  compared.  Since  conjugal  condi- 
tion varies,  as  experience  shows,  with  age,  let  us  divide 
the  two  populations  according  to  age  and  compare  the  cor- 

riage  figure  is  applicable  for  time  comparisons,  in  Die  Grundziige 
der  Theorie  der  Statistik,  pp.  149  and  161. 

""  See  above,  p.  75  f. 

^""aSee  the  admirable  paper  by  G.  Udny  Yule  for  illustration  of 
the  method  of  correcting  crude  birth  rates  to  obtain  birth  rates 
that  take  into  account  both  the  number  of  wives  in  the  population 
and  their  ages  ( "  The  Changes  in  the  Marriage-  and  Birth-Rates 
in  England  and  Wales  during  the  Past  Half  Century;  with  an 
Inquiry  as  to  Their  Probable  Causes,"  Jour,  of  the  Roy.  Stat.  Soc, 
March,  1907 ) .  Also  see  Newsholme  and  Stevenson  on  "  The  Decline 
of  Human  Fertility  in  the  United  Kingdom  and  Other  Countries  as 
Shown  by  Corrected  Birth-Rates"  (Jour,  of  the  Roy.  Stat,  Soc, 
March,  1907 ) . — Teajstslator. 


NATURE  AND  PURPOSE  OF  AVERAGES  109 

responding  age  classes;  it  may  result  from  this  that  in 
each  single  age  class  the  population  B  has  more  widows 
than  A,  though  A  as  a  whole  has  the  larger  quota.  This 
apparent  contradiction  may  easily  arise  from  a  different 
age  division.  The  older  periods  of  life,  of  course,  show 
in  both  populations  many  more  widows  than  the  younger. 
The  older  age  periods  of  A  accordingly  have  many  more 
widows  than  the  younger  age  periods  of  B.  If  now  the 
older  age  periods  in  A  are  relatively  more  populous  than  in 
B,  there  necessarily  results  for  A,  when  all  age  classes  are 
united,  a  larger  percentage  of  widows  than  for  B,  although 
B  in  each  individual  age  class  has  more  widows  than  A, 

In  order  to  make  clear  the  difficulty  of  the  comparison 
of  coordinate  numbers  for  non-homogeneous  masses,  we  may 
refer  to  the  best  known  case,  that  of  the  general  death  rate. 
Let  us,  for  the  sake  of  simplicity,  consider  merely  the  differ- 
ence of  mortality  according  to  age.  If  we  try  to  compare 
the  mortality  of  two  countries  by  means  of  the  general 
death  rate,  it  may  happen  that  country  A  has  the  higher 
general  rate  than  country  B,  although  in  B  the  individual 
age  classes  in  themselves  may  show  the  higher  mortality. 
This  apparently  contradictory  condition  would  occur  if, 
for  instance,  in  A  those  age  classes,  which  (like  the  early 
years  of  infancy)  naturally  have  a  greater  mortality,  were 
relatively  stronger  than  in  B.^"^^  Thus,  we  may  not  without 
further  investigation  attribute  a  higher  general  death  rate 
to  less  favorable  mortality  conditions.  To  make  the  general 
death  rates  of  different  countries  comparable  is  the  purpose 
of  the  method  of  mortality  index,  which  has  given  rise  to 
much  discussion.^"^ 

^''b  Westergaard  gives  an  illustration  of  the  same  apparently  con- 
tradictory condition  as  that  cited  above  in  his  Mortalitiit  und 
Morbilitiit;  the  general  death  rate  of  clergjTnen  is  greater  than  that 
of  railway  employees;  but  the  death  rates  of  clergymen  by  age 
groups  are  much  lower  than  the  corresponding  rates  of  railway 
employees. — Translator. 

^^  Cf.  below,  p.  160  f. 


110  STATISTICAL  AVERAGES  IN  GENERAL 

4.   INVESTIGATION  OF  CAUSATION  BY  COMPARISON  OF  AVERAGES 
AND  RELATIVE  NUMBERS 

The  comparison  of  averages  and  relative  numbers  pos- 
sesses an  especial  importance  as  a  method  of  investigating 
causes,  where  from  the  difference  in  numerical  value  of 
two  or  more  averages  or  relative  numbers  we  infer  a  differ- 
ence of  causation  in  the  phenomena  thus  characterized. 

If  we  compare  averages  or  relative  numbers  which  refer 
to  different  concrete  geographical  districts  or  time  periods, 
and  if  we  establish  a  considerable  difference  in  numerical 
value,  we  perceive  from  this  divergency  that  influences, 
either  quite  different  or  of  varying  strength,  have  affected 
the  masses  compared;  but  we  have  no  further  information 
about  the  difference  which  exists  between  the  causes  operat- 
ing upon  the  two  masses,  and  we  do  not  learn  to  what 
influence  to  attribute  the  difference  of  the  averages  or 
relative  numbers.  If,  for  example,  we  ascertain  that  in 
country  B  the  average  length  of  life  is  shorter,  and  the 
death  rate  higher  than  in  A,  or  if  we  find  that  in  the 
same  country  the  average  length  of  life  has  increased  and 
the  death  rate  fallen  from  decade  to  decade,  this  proves 
indeed  that  there  exists  between  the  masses  compared  a 
difference  in  regard  to  their  causation,  but  we  cannot  infer 
what  the  nature  of  this  difference  is.  "We  must  try  to 
answer  this  question  by  further  statistical  methods  or  in 
some  non-statistical  way. 

The  case  is  different,  however,  when  two  masses  are 
compared  which  are  distinguished  from  each  other  by  a 
definite  qualitative  or  quantitative  criterion  (for  instance, 
sex,  occupation,  age,  etc. ) ,  or  by  an  abstract  space  criterion 
(altitude,  temperature,  soil,  etc.),  or  by  an  abstract  time 
criterion  (season,  etc.).  If  in  such  a  case  a  considerable 
difference  is  established  between  the  averages  or  relative 
numbers  computed  for  the  masses  in  question,  it  is  per- 
missible, under  definite  conditions  to  be  discussed  later, 


NATURE  AND  PURPOSE  OF  AVERAGES  111 

to  attribute  this  difference  to  the  difference  of  criterion  in 
those  masses.  Thus,  if  men  and  women,  or  different 
occupations,  or  age  classes  show  a  varying  mortality  or 
different  length  of  life,  we  may,  under  definite  conditions, 
conclude  that  sex,  or  occupation,  or  age  influences  mortality 
or  length  of  life. 

This  is  the  method  which  corresponds  in  statistics  to 
that  inductive  method  w^hich  J.  S.  Mill  in  his  Logic  called 
the  "  method  of  difference."  If  we  compare  two  groups 
of  phenomena  which  differ  from  each  other  in  one  respect 
both  in  antecedent  and  consequent,  we  conclude,  according 
to  this  method,  that  the  members  in  which  the  compared 
sequences  differ  stand  to  one  another  in  causal  relation. 
If  one  group  consist  of  antecedents  A  B  C  D  and  of  con- 
sequents abed,  and  a  second  group  of  antecedents  BCD 
and  consequents  b  c  d,  it  follows  that  A  is  the  cause  of  a. 
Statistics  has  a  similar  problem  in  the  comparison  of 
averages  and  relative  numbers  for  statistical  masses  which 
differ  in  regard  to  a  definite  objective  factor.  Thus,  from 
the  fact  that  two  statistical  masses  differ,  on  the  one  hand, 
in  regard  to  the  sex  of  the  individuals  in  question  and, 
on  the  other  hand,  in  regard  to  mortality,  w^e  conclude 
that  sex  influences  mortality.  But  there  are  essential  dif- 
ferences between  statistical  material  and  the  material  of 
the  natural  sciences,  to  which  the  inductive  method  is 
regularly  applied.  From  a  conclusion  in  natural  science 
a  general  law  is  obtained  which  admits  of  no  exception. 
Such  a  conclusion  becomes  invalid  if  a  single  case  is  known 
which  contradicts  it.  If  we  have  observed  that  several 
individuals  of  the  same  species  possess  a  definite  mark  or 
character,  and  if  we  infer  from  the  given  evidence  that  the 
whole  species  possesses  this  mark,  this  conclusion  is  in- 
validated by  a  single  instance  to  the  contrary.  It  is  quite 
different  in  regard  to  statistical  conclusions.  These  never 
hold  good  for  single  cases,  but  only  for  statistical  masses 
(aggregates).     The  proposition,  that  the  average  length  of 


112  STATISTICAL  AVERAGES  IN  GENERAL 

life  of  men  is  shorter  than  that  of  women,  hy  no  means 
asserts  that  all  men  die  younger  than  women,  but  only 
that  on  the  whole,  on  the  average,  women  reach  a  greater 
age.  This  proposition  is  wholly  compatible  with  an  ex- 
tremely large  number  of  opposite  cases.  Therefore,  this 
method  of  causal  investigation  by  comparison  of  averages 
and  relative  numbers  cannot  be  regarded  as  the  usual  in- 
ductive process — as  the  majority  of  statisticians  seem  to 
regard  it — even  though  it  is  very  closely  related  to  this 
process.  We  must,  rather,  agree  with  A.  A.  Tschuprow,  who 
has  demonstrated  in  his  very  instructive  article,  "  Die 
Aufgaben  der  Theorie  der  Statistik,"  ^°^  that  the  statistical 
method  should  be  given  an  equal  place  with  the  inductive 
process  in  the  system  of  formal  logic.  Both  methods,  ac- 
cording to  Tschuprow,  have  the  same  purpose — to  establish 
"  natural  laws  "  by  refining  the  raw  material  of  observa- 
tion; but  they  are  applied  under  different  conditions;  the 
inductive  methods  serve  to  disclose  an  invariable  connection 
between  cause  and  effect;  the  statistical  method,  on  the 
other  hand,  is  the  essence  of  such  methods  of  investigation 
as  render  possible  the  study  of  the  looser  causal  connections 
characterized  by  the  plurality  of  causes  and  effects.^"^^ 

In  order  to  infer  a  difference  in  cause  from  the  diver- 
gency of  the  numerical  values  of  two  averages  or  relative 

^"*  Jahrbuch  fiir  Gesetzgebung,  Verwaltung  und  Volkswirtschaft 
im  Deutsehen  Reiche.  Edited  by  Gustav  Schmoller.  Vol.  XXIX, 
Pt.  II,  1905,  p.  27. 

'"SaFor  discussions  of  the  scope  and  methods  of  economics  and 
statistics  see  Venn's  Logic  of  Chance,  Ernst  G.  F.  Gryzanovski's 
paper  "  On  Collective  Phenomena  and  the  Scientific  Value  of  Sta- 
tistical Data"  (published  as  No.  3,  Vol.  VII,  Third  Series,  of  the 
Publications  of  the  Am.  Econ.  Assoc,  August,  1906),  H.  L.  Moore's 
article  on  "  The  Statistical  Complement  of  Pure  Economics "  in  the 
Quar.  Jour.  Eeons.,  November,  1908,  also  his  Laws  of  Wages  (Mac- 
millan,  1911),  Keynes'  Scope  and  Method  of  Political  Economy,  and 
the  article  on  "  Method  of  Political  Economy,"  in  Palgrave'a  Diet. 
Pol.  Econ. — Teanslatob. 


NATURE  AND  PURPOSE  OF  AVERAGES    U3 

numbers,  the  divergency  must  be  considerable  and  impor- 
tant. Minor  differences  will  naturally  not  justify  such  an 
inference.  The  statistician  will  have  to  judge  in  the  in- 
dividual case  if  a  difference  is  significant.  For  the  ele- 
mentary mathematical  statistician  the  decision  is  more  or 
less  a  question  of  subjective  estimate.  The  calculus  of 
probability  makes  it  possible  for  the  mathematical  statis- 
tician, on  the  other  hand,  to  determine  in  many  cases  with 
what  probability  the  difference  between  the  two  values 
compared  may  be  regarded  as  accidental,  or  what  probabil- 
ity there  is  for  the  existence  of  a  different  causation. 

The  number  of  objective  criteria,  according  to  which 
statistical  masses  may  be  differentiated,  is  known  to  be  ex- 
tremely large,  and  in  very  many  cases  the  masses,  which 
differ  in  regard  to  a  definite  criterion,  produce  in  fact 
averages  and  relative  numbers  of  different  numerical  values. 
For  instance,  the  death  rate  is  different  for  each  of  the 
sexes,  for  various  age  classes,  occupations,  etc. ;  it  varies 
with  the  season  and,  apparently,  with  the  altitude.  Simi- 
larly, the  average  length  of  life,  the  average  age  at  mar- 
riage, the  marital  fecundity  of  different  groups  of  the  popu- 
lation differ  from  one  another.  Various  phenomena  of 
moral  and  economic  statistics  also  show  characteristic  dif- 
ferences for  different  groups  of  population.  The  deter- 
mination of  such  divergencies  is  one  of  the  most  important 
tasks  of  statistics,  and  statisticians  must  continually  en- 
deavor to  disclose  new  characteristic  differences  by  an  ever 
increasing  differentiation  of  statistical  material.  Often  in 
the  statistical  investigation  of  causes  a  causal  connection 
is  presumed  on  the  basis  of  some  extraneous  knowledge. 
This  connection  is  then  to  be  statistically  proved.  In  this 
case  an  hypothesis  is  first  set  up  and  then  verified  by  a 
tentative  division  of  the  statistical  material  (by  an  "  ex- 
perimental formation  of  groups  ")  and  by  the  comparison 
of  parts  which  are  distinguished  from  each  other  in  regard 
to  the  factor  which  is  considered  causal. 


114  STATISTICAL  AVERAGES  IN  GENERAL 

But  the  statistical  method  is  only  able  to  prove  causality 
within  definite  limits.  The  comparison  of  averages  or  rela- 
tive numbers  may  show  that  definitely  characterized  masses, 
which  differ  from  each  other  in  certain  respects,  yield  rela- 
tive numbers  or  averages  of  different  numerical  values.  The 
result  of  the  comparison  is  a  coincidence  of  definite  facts 
which,  indeed,  allows  us  to  infer  a  causal  connection,  but 
which  gives  no  information  as  to  which  fact  is  the  cause 
and  which  the  effect.  This  question  must  be  decided  in 
the  individual  case  on  the  basis  of  further  knowledge, 
perhaps  by  means  of  further  statistical  investigations.^"^*' 

The  decision  will  generally  be  very  easy.     Thus,  if  we 

^°'b  Concerning  this  point  Prof.  H.  L.  Moore  holds  that  "  Eco- 
nomic events  are  not  arrayed  in  linear  connection,  the  one  event 
following  the  other  in  direct  series,  as  was  frequently  assumed 
by  the  classical  economists.  It  was  an  idle  controversy  that  Mal- 
thus  and  Ricardo  conducted  upon  the  question  whether  the  abundance 
of  food  increases  the  population  or  the  multitude  of  consumers  in- 
creases the  supply  of  food.  Social  phenomena  are  interrelated,  are 
mutually  dependent,  and  the  appropriate  method  of  treating  such 
a  form  of  interdependence  is  the  use  of  a  system  of  simultaneous  equa- 
tions in  which  the  equations  are  equal  in  number  to  the  unknown 
quantities  in  the  problem  "  ( Laws  of  Wages,  p.  2 ) .  However,  there 
is  a  controversy  going  on  at  the  present  time  that  appears  to  hinge 
on  just  this  question  of  the  order  of  economic  phenomena.  It  is 
the  question  of  the  relation  between  the  quantity  of  money  and 
prices.  J.  L.  Laughlin  is  the  leader  of  the  school  holding  that  varia- 
tions in  prices  precede  and  cause  variations  in  the  amount  of  cur- 
rency. He  says,  "  When  the  price  is  fixed,  the  credit  medium  by 
which  the  commodity  is  passed  from  seller  to  buyer  comes  easily 
and  naturally  into  existence  and,  of  course,  for  a  sum  exactly  equal- 
ing the  price  agreed  upon  multiplied  by  the  number  of  units  of 
goods.  .  .  .  That  is,  the  quantity  of  the  actual  media  of  exchange 
thus  brought  into  use  is  a  result  and  not  a  cause  of  the  price-making 
process"  (Bui.  Am.  Econ.  Assoc,  April,  1911,  pp.  28,  29).  The 
classical  theory  is  that  variations  in  the  quantity  of  the  media  of 
exchange  precede  and  cause  variations  in  the  price-level.  Irving 
Fisher  is  the  chief  modern  protagonist  of  this  theory.  He  holds  that 
the  price-level  "is  not  cause  but  effect"  (Bui.  Am.  Econ.  Assoc, 
April,  1911,  p.  38). — Tbanslatob. 


NATURE  AND  PURPOSE  OF  AVERAGES  115 

find  that  the  more  wealthy  classes  of  the  population  show 
a  lower  mortality,  we  may  evidently  designate  the  wealth 
as  cause  and  the  lower  mortality  as  effect.  The  case  will 
not  be  so  simple  if  we  determine,  for  instance,  a  coincidence 
of  greater  wealth  with  fewer  children.  Is  the  wealth  the 
cause  of  the  small  number  of  children,  as  is  often  asserted, 
or  is  it  not  also  conceivable  that  many  families  have  attained 
to  wealth  by  reason  of  the  smaller  number  of  children? 
Another  interesting  example  is  the  coincidence  of  greater 
infant  mortality  with  higher  birth  rate.  It  seemed  self- 
evident  that,  in  general,  the  reason  was  that  as  the  number 
of  children  per  family  increased  the  care  and  attention 
bestowed  on  each  must  necessarily  decrease.  But  the  very 
opposite  causal  connection  has  been  proved,  at  least  for  a 
certain  group  of  cases.  Geissler  has  ascertained  ^°^  the 
period  clasping  between  the  births  of  two  successive  chil- 
dren in  the  same  family  for  26,429  families  of  Saxon 
miners,  and  he  has  differentiated  these  measurements  ac- 
cording to  whether  the  firstborn  child  died  or  remained 
alive.  He  has  demonstrated  that  the  interval  was  shorter 
if  the  older  child  died.  Evidently,  if  a  child  died,  the 
desire  was  aroused  for  another  child,  or  else  the  checks, 
which  usually  delayed  the  begetting  of  another  child, 
vanished.  This  fact  shows  that  not  only  the  number  of 
children  may  influence  mortality  but  also,  under  certain 
circumstances,  that  mortality  may  influence  the  number 
of  children.  Here  is  evidently  a  case  of  mutual  influence, 
of  interdependence,  such  as  the  social  organism  so  often 
shows.  In  many  other  cases  there  is  no  immediate  causal 
connection  at  all  between  the  two  facts  whose  coincidence 
is  statistically  shown ;  they  are  both  under  the  influence 
of  a  deeper  common  cause.  Thus,  if  we  find  that  criminals 
are  on  the  average  smaller  than  non-criminals,  a  direct 
connection  between  bodily  size  and  criminality  is.  naturally, 
excluded,  but  deeper  lying  common  causes  may  exist  to 
109  Zeitschrift  des.  konigl.  sachs.  statistischen  Bureaus  1885,  p.  24. 


116  STATISTICAL  AVERAGES  IN  GENERAL 

which  physical  inferiority  and  moral  depravity  may  be 
ultimately  attributed. 

When  we  speak  in  statistics  of  the  investigation  of  causes, 
we  are,  of  course,  never  concerned  with  investigating  the 
causes  of  single  cases  or  individual  events.  The  statistical 
investigation  of  causes  can  refer  only  to  characteristic  con- 
ditions, in  which  masses  of  individuals  (or  other  units) 
are  found.  These  conditions  are  often,  so  to  speak,  merely 
the  frame  in  which  there  are  operative  individual  causes 
of  various  kinds  and  intensities,  about  which  the  statistical 
comparison  itself  yields  no  information.  As  regards  these 
individual  causes,  other  sciences,  according  to  the  object 
of  the  investigation,  may  supply  information ;  so  far  as 
they  affect  a  considerable  number  of  individuals  they  may 
be  determined  by  means  of  further  detailed  statistical  in- 
vestigations;  but  they  may  also  remain  quite  unknown. 
Thus,  the  comparison  of  the  mortality  of  boys  and  girls 
or  of  legitimate  and  illegitimate  children  develops  the  un- 
doubted fact  that  there  is  a  greater  mortality  among  the 
boys  and  among  the  illegitimate  children ;  popularly  speak- 
ing, therefore,  membership  in  the  male  sex  and  illegitimacy 
are  designated  as  the  causes  of  the  greater  mortality.  The 
real  immediate  causes  which  result  in  the  individual  deaths 
of  boys  more  than  of  girls  and  of  illegitimate  more  than 
of  legitimate  children — are  not  disclosed  by  the  statistical 
comparison;  they  can  only  be  discovered  by  medical  re- 
search or  by  further  statistical  investigation  of  details.  The 
same  thing  is  true  if  we  follow  the  influence  of  wealth  upon 
various  demographic  phenomena,  for  instance,  on  mortality. 
To  ascertain  that  poverty  increases  mortality  does  not  dis- 
close the  causes  immediately  operative.  But  by  further 
and  more  special  investigations  we  may  inquire  in  what 
way  wealth  affects  the  causes  of  death,  whether  the  same 
diseases  take  a  different  course  with  the  wealthy  and  the 
poor,  etc. 

The  situation  is  the  same  in  numerous  other  cases;  for 


NATURE  AND  PURPOSE  OF  AVERAGES  117 

example,  if  we  measure  the  influence  of  conjugal  condition, 
occupation,  the  seasons,  etc.,  on  various  demographic,  moral- 
statistical,  and  economic  phenomena.  Even  though  in  all 
these  cases  the  direct  individual  causes  are  not  discovered 
by  the  statistical  method  in  question  and  even  though  we 
may  not  speak  of  the  determination  of  real,  exact,  causal 
laws,  yet  by  this  method  regularly  recurring  differences  are 
discovered.  These  differences  found  to  exist  between  defi- 
nitely characterized  groups  of  individuals  (or  other  units) 
cannot  be  disclosed  by  other  methods  than  the  statistical, 
and  their  discovery  forms  the  starting  point  of  further  and 
more  thorough  investigations  and,  consequently,  may  also 
furnish  a  basis  for  measures  of  social  reform.^^"'^^^ 

If  the  difference  between  two  averages  or  relative  num- 
bers is  to  be  causally  related  to  a  precise  difference  in 
the  masses  compared,  then  these  masses  must  be  assumed 
to  differ  from  each  other  only  in  that  precise  way,  but 
to  agree  in  all  other  respects.  The  conclusion  that  there  is 
a  causal  connection  is  only  permissible  on  the  assumption 
that  other  things  are  equal.  If  this  assumption  is  not 
true,  if  the  masses  diverge  also  in  regard  to  another  cri- 
terion besides  the  one  used  to  differentiate  them,  then  we 

"°  Interesting  examples  are  the  investigations  as  to  the  influence 
of  the  various  occupations  on  mortality  and  on  diseases,  and  the 
attempts  to  get  at  the  actual  causes  such  as  stooping  posture,  dust, 
fumes,  dampness,  etc. 

^^^  While  there  are  plausible  explanations  for  most  statistically 
determined  differences  (such  as  the  influence  of  economic  position, 
of  the  seasons,  etc.)  the  fact  of  the  different  sex- ratio  in  living  births 
and  still-births,  in  legitimate  and  illegitimate  children,  seems  quite 
inexplicable  to  the  layman.  Lexis  ( Abhandlungen  zur  Theorie  der 
Bevolkerungs-  imd  Moralstatistik,  VII,  "  Das  Geschlechtsvcrluiltnis 
der  Geborenen  und  die  Wahrscheinlichkeitsrechnung,"  p.  166  fF.)  has 
tried  to  relate  these  differences  to  different  percentages  of  early 
births;  G.  v,  Mayr  (Bevolkerungsstatistik,  p.  188)  has  connected 
the  greater  excess  of  boys  in  the  country  as  compared  with  the  city 
with  the  relatively  greater  inbreeding  and  has  explained  in  the 
same  way  the  excess  of  boys  among  the  Jews. 


118  STATISTICAL  AVERAGES  IN  GENERAL 

cannot  decide  what  difference  in  the  masses  is  the  cause 
of  the  divergence  of  the  averages  or  relative  numbers. 

If  we  find,  for  instance,  that  the  male  and  female  labor- 
ers of  a  definite  district  have  different  average  wages,  but 
if  we  know  at  the  same  time  that  the  men  are  in  different 
occupations  than  the  women,  we  cannot  decide  whether  the 
difference  in  the  average  wages  is  to  be  attributed  to 
difference  of  sex  or  to  difference  in  the  categories  of  work. 
If  we  compare  the  mortality  in  various  occupations  and 
know  that  those  belonging  to  these  occupations  have  a  differ- 
ent age  grouping,  we  are  not  justified  in  attributing  the 
difference  in  mortality  to  the  occupation,  since  it  may  also 
come  from  the  different  age  grouping. 

The  prerequisite  that  other  things  be  equal  is  seldom 
completely  fulfilled;  a  certainty  that  it  is  fulfilled  can 
never  be  attained.  The  conclusion  of  causal  connection 
will,  therefore,  always  be  merely  hypothetical  and  more  or 
less  probable.^^^-"^* 

*^^v.  Inama-Sternegg  ("Neue  Beitrage  zur  Methodenlehre  der 
Statistik "  in  Staatswissenschaftliche  Abhandlungen,  1903)  dis- 
tinguishes the  progressive  method  (experiment),  by  which  we  pro- 
ceed from  cause  to  effect,  and  the  regressive  method  (observation), 
by  which  we  infer  the  cause  from  the  effect.  He  asserts  that  while 
the  direct  proof  of  a  causal  connection  is  obtained  by  the  progressive 
method  the  regressive  method  leads  only  to  an  explanation  that  is  a 
mere  hypothesis. 

Although  the  great  majority  of  statisticians  recognize  in  the 
investigation  of  causality  one  of  the  most  important,  though  most 
diflBcult  tasks  of  their  science,  theoretical  opponents  have  not  been 
wanting.  Napoleone  Colajanni  (Statistica  teorica,  p.  265)  mentions 
Bodio  as  such.  G.  Staehr  ("  Einige  Bemerkungen  iiber  die  statistische 
Methode,"  Bulletin  de  I'lnstitut  international  de  Statistique,  Vol.  IV, 
No.  1,  Pt.  II,  p.  288  ff.)  denies  that  statistics  is  competent  to  deter- 
mine causes;  he  thinks  it  is  an  empirical-descriptive  but  not  an 
inductive-analytical  science. 

112a  Prof.  H.  L.  Moore  holds  that  the  argument  of  statistics 
is  purely  utilitarian  or  pragmatic  in  character.  He  quotes  from 
Jevons's  Theory  of  Political  Economy  as  follows:  "The  deductive 
science  of  economics  must  be  verified  and  rendered  useful  by  the 


NATURE  AND  PURPOSE  OF  AVERAGES    119 

The  assumption  that  other  things  are  equal  will  possess 
the  greatest  probability  of  being  true  in  comparing  highly- 
homogeneous  masses,  for  instance,  if  we  compare  the  aver- 
age wages  of  male  laborers,  on  the  one  hand,  and  female 
laborers,  on  the  other  hand,  of  the  same  occupation  and 
of  a  single  category  of  work,  or  the  mortality  of  individuals 
of  the  same  sex  and  age  of  two  different  occupations.  But, 
as  already  emphasized,  complete  homogeneity  is  never 
reached.  On  the  contrary,  statistics  is  regularly  concerned 
with  masses  which  cannot  be  called  homogeneous.  Two 
non-homogeneous  masses  differ  solely  with  respect  to  the 
criterion  by  which  they  are  differentiated  (the  condition  of 
a  conclusion)  only  in  case  they  are  made  up  in  a  like  man- 
ner of  groups  of  the  more  homogeneous  constituents,  the 
groups  corresponding  to  the  other  criteria  coming  into  con- 
sideration ;  only  in  such  a  case  can  the  comparison  of  non- 
homogeneous  masses  lead  to  the  isolation  and  measurement 
of  a  definite  influence.  For  instance,  if  we  compare  the 
average  wages  of  all  the  male  and  female  laborers  of  a 
definite  district,  a  conclusion  as  to  the  causal  influence  of 
sex  upon  wages  will  only  be  possible  if  it  is  certain  that 
the  male  and  female  laborers  are  similarly  distributed  in 
the  different  occupations  and  categories  of  work,  and  that 
they  possess  the  same  age  classification,  etc. ;  from  the  com- 
parison of  the  mortality  of  those  belonging  to  different 
occupations,  the  influence  of  occupation  will  only  be  mani- 
fest if  the  sex  and  age  classification,  etc.,  of  the  persons 
compared  is  the  same.  The  investigation  of  causality  by 
a  comparison  of  statistical  averages  and  relative  numbers 
is  thus  limited  by  prerequisites  which  must  be  strictly 
examined  and  which,  unfortunately,  are  often  not  fulfilled. 
But  statistics  has  to  depend  on  the  material  at  its  disposal 
and  cannot  like  physics,  for  instance,  devise  experiments 

purely  empirical  science  of  statistics."  ( See  "  The  Statistical 
Complement  of  Pure  Economics,"  Quarterly  Journal  of  Economics, 
November,  1908,  p.  16.) — Tbanslatob. 


120  STATISTICAL  AVERAGES  IN  GENERAL 

in  order  to  observe  and  measure  the  effect  of  a  factor  which 
may  be  added  or  eliminated  at  pleasure.  The  statistical 
investigation  of  causes,  therefore,  gives  rise  to  errors  only 
too  frequently,  and  a  perfect  theoretical  certainty  can  never 
be  obtained. 

The  conscientious  statistician  must,  accordingly,  renounce 
entirely  any  conclusion  as  to  causality  if  the  masses  com- 
pared by  him  differ  not  only  in  regard  to  the  one  criterion 
employed  but  also  exhibit  other  differences  whose  influence 
cannot  be  accounted  for  numerically  or  be  ignored  as 
trifling.  Influences  impossible  to  estimate  occur,  unfortu- 
nately, only  too  often.  Thus,  a  proof  of  the  influence  of 
wealth  upon  mortality  is  generally  impossible  because  the 
members  of  the  different  economic  classes  belong  to  various 
occupations.  We  encounter  a  similar  difficulty  if  we  try 
to  determine  the  influence  of  various  religions  on  the 
morality  of  the  population,  since  religious  differences  gen- 
erally coincide  with  national  differences. 

Mention  should  also  be  made  of  the  controversy  which 
was  once  waged  in  regard  to  the  influence  of  conjugal 
condition  on  mortality.  From  the  fact  of  the  lower  mor- 
tality of  married  people,  several  authors  (among  them 
Bertillon)  concluded  that  the  influence  of  marriage  was 
beneficial.  Block  opposed  this  view,  pointing  to  the  factor 
of  selection  and  showing  that  many  people  do  not  marry 
on  account  of  physical  ailments  and  that  those  who  do 
marry  are  accordingly  healthier  than  those  who  do  not 
marry;  therefore,  lower  mortality  for  the  married  should 
be  expected  a  priori.  A  strict  statistical  proof  would  only 
be  obtained  if,  on  the  one  hand,  married,  and  on  the  other 
hand,  unmarried  people  of  equal  physical  condition  (and 
of  the  same  occupations,  wealth,  etc.)  could  be  compared 
in  regard  to  their  mortality, — a  thing  which  is  impossible. 

The  factor  of  selection  is  important  in  many  other  fields, 
as,  for  example,  in  the  comparison  of  the  demographic 
conditions  of  various  occupations.     The  choice  of  an  occu- 


NATURE  AND  PURPOSE  OF  AVERAGES    121 

pation  is  often  made  ])y  a  kind  of  natural  selection,  since 
certain  occupations  require  definite  physical  conditions. 
Hence,  we  cannot  conclude  from  the  different  mortality  of 
those  belonging  to  the  occupations  that  the  occupation  is 
the  cause  of  the  difference. 

In  the  above  cases  we  have  dealt  with  the  comparison 
of  masses  diverging  in  regard  to  a  quantitative  or  qualita- 
tive criterion  which  must  have  been  fixed  when  the  statistics 
were  obtained.  For  instance,  if  the  wages  of  men  and 
women  are  compared,  the  sex  of  the  individual  laborers 
must  have  been  indicated  when  the  wage  data  were  obtained, 
so  that  these  wage  data  could  be  divided  subsequently, 
into  two  masses  according  to  the  sex  of  the  laborers.  The 
case  is  somewhat  different,  when,  in  order  to  establish  a 
causality,  averages  of  time  or  place  series  (or  parts  of 
them)  are  compared,  which  series  at  the  same  time  differ 
from  each  other  in  a  quantitative  or  qualitative  respect. 
The  series  to  be  compared  are,  in  this  case,  not  differen- 
tiated according  to  a  factor  considered  when  the  data  were 
obtained,  but  according  to  either  a  non-statistical  criterion 
or  one  taken  from  some  other  statistical  data.  Thus,  we 
may  compare  the  years  before  and  after  a  definite  fact, 
for  instance,  the  promulgation  of  a  new  law,  in  order  to 
see  whether  this  fact  had  an  influence  on  a  definite  phe- 
nomenon and,  if  so,  of  what  importance  it  was.^^^^  Or,  we 
may  compare  periods  of  time  or  districts  which  differ  in 
regard  to  their  economic  condition,  to  which  evidently 
some  causal  significance  belongs;  for  example,  we  may 
compare  periods  of  economic  prosperity  and  depression  in 
regard  to  unemployment,  mortality,  criminality,  etc. 

The  considerations  indicated  above  for  the  comparison 
of  masses  differentiated  in  a  merely  quantitative  or  quali- 

"°bA  fine  illustration  of  such  a  comparison  may  be  found  in  the 
article  by  G.  H.  Wood  on  "  Factory  Legislation  Considered  with 
Keference  to  the  Wages,  etc.,  of  the  Operatives  Protected  Thereby " 
(Jour.  Roy.  Stat.  Soc,  Vol.  LXV,  pp.  284-324 )  .—Tbanslatob. 


122         STATISTICAL  AVERAGES  IN  GENERAL 

tative  way  are  also  true  for  the  comparison  of  time  and 
place  masses  with  simultaneous  qualitative  or  quantitative 
differences;  for  instance,  the  consideration  that  the  statis- 
tical comparison  establishes  the  existence  of  a  causal  con- 
nection but  gives  no  certainty  as  to  which  fact  is  cause 
and  which  is  effect,  holds  true  here.  Furthermore,  time 
and  place  comparisons  of  the  kind  in  question  can  only 
lead,  like  purely  quantitative  and  qualitative  comparisons, 
to  a  conclusion  of  causality  on  the  assumption  that  other 
things  are  equal,  and  in  the  concrete  case  we  must  inquire 
whether  this  assumption  is  permissible. 


C.     AVERAGES  AS  STANDARDS  FOR  JUDGING 
ITEMS 

The  items  of  a  statistical  series  are  often  comparea  with 
its  average  in  order  to  ascertain  whether  particular  items 
are  above  or  below  the  average  and  how  far  they  diverge 
from  it.  This  determination  may  be  of  great  significance 
for  the  judgment  of  the  items,  since  great  deviations  from 
the  average  indicate,  as  a  rule,  the  existence  of  special 
causes.  The  information,  which  may  be  obtained  by  the 
comparisons  of  items  with  the  average,  varies,  however,  ac- 
cording to  the  kind  of  series  involved. 

If  in  a  series  of  quantitative  single  observations  an  item 
differing  greatly  from  the  average  is  found  (for  instance, 
the  wages  or  the  length  of  life  of  a  certain  individual),  it 
is  certain  that  this  difference  is  to  be  attributed  to  special 
causes,  but  the  statistical  method  in  question  is  unable  to 
reveal  the  nature  of  these  special  causes.  If  we  have  a 
series  of  the  second  group,  whose  members  indicate  the 
size  of  definitely  limited  masses,  or  those  series  of  the  third 
group  which  consist  of  relative  numbers  or  averages  re- 
ferring to  different  time  or  place  masses,  it  is  also  impos- 
sible to  determine  a  definite  causal  connection  by  com- 
paring an  item  with  the  average.     Thus,  if  a  definite  mem- 


NATURE  AND  PURPOSE  OF  AVERAGES    123 

ber  of  a  time  series  consisting  of  absolute  or  relative  num- 
bers or  averages  shows  a  striking  divergence  from  the 
arithmetic  mean,  this  will  indeed  indicate  that  special 
causes  have  affected  the  mass  at  the  time  of  the  item  in 
question.  But  these  special  causes  can  only  be  discovered 
by  means  of  other  statistical  investigations  or  in  a  non- 
statistical  way. 

The  case  is  different  when  an  item  of  a  qualitative  or 
quantitative  series  of  the  third  group  is  compared  with 
the  average  of  the  series.  The  item  refers,  in  such  a  case, 
to  a  part  which  differs  from  the  totality  in  regard  to  a 
definite  qualitative  or  quantitative  criterion.  If,  in  such 
a  comparison,  a  considerable  difference  appears  between  the 
relative  number  (or  average)  referring  to  the  part  and 
the  relative  number  (or  average)  characterizing  the  total- 
ity, then  we  may  infer — other  things  being  equal — that  the 
factor  which  especially  characterizes  the  part  is  the  cause 
of  the  difference  in  the  values  compared.  For  example, 
if  we  find  that  those  of  a  certain  occupation  have  a  mor- 
tality considerably  above  the  average,  we  attribute  this 
divergence  from  the  average — other  things  being  equal — 
to  that  occupation;  if  we  find  that  a  disease  appears  more 
frequently  in  a  certain  age  class  than  on  the  average  for 
the  whole  population,  we  ascribe — other  things  being  equal 
— to  the  age  in  question  an  influence  on  the  frequency  of 
the  disease.  This  is  simply  a  variety  of  the  investigation 
of  causality  discussed  in  the  preceding  chapter  by  com- 
paring averages  and  relative  numbers;  this  variety  is 
marked  by  the  fact  that  the  values  compared  are  not  co- 
ordinate— as,  for  instance,  the  death  rate  of  men  as  com- 
pared with  that  of  women — but  are  in  the  mutual  relation 
of  item  and  average.  The  principles  established  in  the  pre- 
ceding chapter  may  therefore  be  applied,  with  the  neces- 
sary changes,  to  the  comparison  of  the  relative  numbers 
and  averages  here  involved. 

The  comparison  of  a  specially  characterized  part  with 


124  STATISTICAL  AVERAGES  IN  GENERAL 

the  totality  is  often  made  in  order  to  ascertain  whether  a 
certain  factor  exerts  an  influence  in  one  way  or  another. 
But  this  comparison  does  not  give  directly  the  precise 
extent  of  this  influence,  for  the  following  reason:  The 
totality  includes  also  the  specially  characterized  part;  the 
particular  influence  to  be  measured  has,  therefore,  affected 
the  numerical  value  for  the  totality  and  hence  does  not 
find  precise  expression  in  the  comparison.  The  smaller 
the  part  in  relation  to  the  totality,  the  less  important  will 
be  the  disturbance.  But  if  the  part  is  a  large  percent  of 
the  totality,  the  value  for  the  totality  is  already  con- 
siderably influenced  and  the  comparison  becomes  of  little 
or  no  significance.  Thus,  the  average  height  of  criminals 
is  often  compared  with  the  average  height  of  the  total 
population.  In  order  to  measure  exactly  the  connection 
of  criminality  with  height,  we  should  really  have  to  com- 
pare not  criminals  and  the  total  population  but  criminals 
and  non-criminals.  If  criminals,  as  it  appears,  are  on  the 
average  smaller  than  the  total  population,  the  non-criminals 
must  on  the  average  be  taller  than  the  total  population; 
accordingly  the  difference  between  criminals  and  non- 
criminals  must  be  greater  than  the  difference  between  crim- 
inals and  the  total  population.  But  since  criminals  are 
only  a  very  small  fraction  of  the  total  population,  the 
average  height  of  the  total  population  is  only  imperceptibly 
influenced  by  them,  and  the  comparison  of  the  part  (crim- 
inals) with  the  totality  (whole  population)  is  sufficient. 
It  would  be  quite  different  if,  for  instance,  we  were  con- 
cerned with  the  influence  of  sex  on  height.  No  one  would 
think  of  comparing  the  average  height  of  persons  of  one 
sex  with  the  average  height  of  the  total  population;  as 
a  matter  of  course,  the  two  sexes  would  be  compared 
directly  with  each  other. 

In  practical  statistics  there  are,  however,  some  cases 
where  a  statistical  item  is  compared  with  an  average  of 
the  same  logical  content.    In  the  computation  of  the  aver- 


NATURE  AND  PURPOSE  OF  AVERAGES    125 

age  in  such  a  case,  the  item  in  question  is  not  taken  into 
account.  For  example,  the  Austrian  harvest  statistics  gives 
the  crop  yield  of  the  individual  years  as  a  percentage  of 
the  average  of  the  preceding  ten  years  (thus,  the  crop 
of  1904  in  relation  to  the  average  of  the  period  1894-1903). 
Among  the  motives  which  may  lead  to  such  a  proceeding 
there  is  perhaps  the  consideration  mentioned  above,  that 
the  comparison  of  an  item  with  an  average,  whose  size 
has  been  affected  by  the  item  itself,  does  not  clearly  express 
the  strength  of  the  particular  causes  influencing  that  item. 
A  similar  case  often  occurs  when  the  level  of  prices  of 
various  years  is  compared  by  means  of  total  index  numbers. 
A  single  year  is  frequently  not  chosen  as  a  basis  of  com- 
parison, but  the  average  of  a  definite  number  of  years, 
the  years  of  the  '*  standard  period."  ^^^  With  this  average 
are  compared  not  only  the  years  belonging  to  the  standard 
period — in  which  items  are  compared  with  their  average — 
but  also  the  years  preceding  or  following  the  standard 
period. 

D.    THE    FUNCTION    OF    AVERAGES    IN    THE    MEAS- 
UREMENT OF  THE  DISPERSION  OF  SERIES 

Averages  may,  as  we  have  already  explained,  serve  as 
standards  for  the  judgment  of  items.  If  not  merely  a 
single  item  of  a  series  is  judged  by  the  average,  but  rather 
all  the  items  of  the  series,  we  obtain  a  picture  of  the  group- 
ing or  dispersion  of  the  whole  series  about  the  average. 
To  obtain  such  a  picture  is  very  often  necessary  for  the 
judgment  of  statistical  series.  The  dispersion  of  a  series 
of  individual  observations  indicates  the  degree  of  the  varia- 
bility of  an  individual  character  (for  instance,  height, 
wages,  etc.),  whose  measurement  is  very  often  of  the  great- 

"'  Sauerbeck  takes  the  years  1867-1877  as  the  standard  period, 
The  Economist  the  years  1845-1850,  Soetbeer  the  years  1847-1850, 
Conrad  both  the  years  1879-1883  and  the  years  1879-1889,  etc. 


126  STATISTICAL  AVERAGES  IN  GENERAL 

est  significance;  the  dispersion  of  a  time  series  gives  a 
measure  of  the  constancy  or  variability  of  a  phenomenon 
in  the  course  of  time;  the  dispersion  of  a  geographical 
series  reveals  the  variations  to  which  a  phenomenon  is 
subject  in  the  districts  under  consideration,  etc. 

Hence,  averages  are  often  computed  to  serve  as  a  point 
of  departure  for  the  investigation  of  the  grouping  of  the 
items.  This  investigation  is  of  prime  importance  since 
statistical  series  show  the  greatest  multiplicity  in  regard 
to  the  distribution  of  their  items,  and  scarcely  two  series 
exist  of  similar  and  equally  great  dispersion.  This  in- 
vestigation also  makes  it  possible  to  divide  the  statistical 
series  into  different  sub-classes.  Many  statistical  series 
show  a  more  or  less  regular  dispersion.  These  are  the  series 
which  can  be  particularly  well  expressed  by  averages.  The 
averages  from  such  series  may  be  supplemented  by  special 
values,  which  mark  the  dispersion  of  the  series  about  their 
averages.  Those  series  may  be  best  designated  in  this  way 
whose  members  are  symmetrically  grouped  about  the  aver- 
age according  to  the  law  of  chance.  The  indication  of  the 
average  and  of  a  measure  of  dispersion  (such  as  the  average 
deviation  or  the  standard  deviation)  are  enough  in  such 
a  case  to  characterize  the  series  in  its  entirety.  Other 
series  are  not,  indeed,  distributed  symmetrically  about  their 
averages,  but  yet  the  grouping  of  the  items  about  the  aver- 
age may  be  brought  under  an  extended  law  of  chance. 
Series  whose  members  show  no  sort  of  regular  grouping 
about  their  means  may,  moreover,  be  divided  into  more 
homogeneous  parts  possessing  a  regular  dispersion.  Among 
the  series  of  non-symmetrical  dispersion  about  the  average 
those  deserve  a  special  interest  which  show  a  characteristic 
regular  conformation  in  some  other  way. 

In  the  above  we  have  dealt  with  the  case  where  an  aver- 
age is  computed  for  the  purpose  of  serving  as  a  starting 
point  for  the  measurement  and  judgment  of  the  dispersion 
of  the  series;  in  such  a  case  the  measurement  of  the  dis- 


NATURE  AND  PURPOSE  OF  AVERAGES    127 

persion  is  the  real  object,  the  computation  of  the  average 
simply  a  means  to  this  end.  But  the  determination  of 
the  dispersion  is  also  very  useful  where  an  average  is  used 
for  an  independent,  primary  purpose,  such  as  the  compre- 
hensive characterization  of  a  series,  or  for  purposes  of 
comparison.  An  average  in  itself  in  no  way  expresses  the 
dispersion  of  the  series  from  which  it  arises,  and  yet  the 
methodological  value  of  each  average,  especially  the  ques- 
tion whether  it  may  be  regarded  as  a  "  typical  ' '  average  or 
not,  depends  on  this  dispersion.  Therefore,  when  averages 
are  employed,  data  should  also  be  obtained  in  regard  to 
the  dispersion  of  the  series. 

In  measuring  the  dispersion  of  statistical  series  of  in- 
dividual observations,  we  may  start  from  various  averages, 
the  arithmetic  mean,  the  mode,  etc.  The  description  of 
the  various  methods  of  measuring  dispersion  can,  therefore, 
only  follow  the  discussion  of  the  various  kinds  of  averages. 


PART  II 
THE  VAEIOUS  KINDS  OF  AVERAGES 


CHAPTER  I 

SYNOPSIS 

Each  average,  as  has  been  said,  serves  to  characterize 
a  series  by  a  single  numerical  expression.  This  characteri- 
zation can  be  accomplished  in  various  ways  and  conse- 
quently various  kinds  of  averages  are  possible.  A  complete 
enumeration  of  all  the  numerical  values  which  may  char- 
acterize a  series  is,  of  course,  impossible.^ 

Only  those  kinds  of  averages  will  be  considered  in  the 
following  which  are  actually  used  in  statistics.  Such  are 
the  arithmetic  mean,  including  the  weighted  mean,  the 
geometric  mean,  the  median,  and  the  mode.  The  peculiar 
properties  of  each  of  these  means  will  be  investigated,  that 
is,  what  they  indicate  about  the  series  in  question,  how 
they  are  computed,  in  what  departments  of  applied  statis- 
tics they  are  useful,  and  the  like. 

^  Fechner  says  ("Uber  den  Ausgangswert  der  kleinsten  Abweich- 
ungssumme,"  Abhandlungen  der  kgl.  sachsischen  Gesellschaft  der 
Wissenschaften,  Vol.  XVIII,  p.  74)  :  "By  a  mean  of  given  items  we 
understand  a  value  which  can  be  derivpd  from  these  items  according 
to  definite  principles  and  which  falls  among  the  values  of  the  items, 
or  more  shortly,  a  value  which  is  a  function  of  the  items  falling 
between  the  minimum  and  maximum  items.  Consequently,  there  are 
an  indefinite  number  of  means  as  there  are  an  unlimited  number 
of  definite  principles  or  functions  of  the  kind  described.  Only  those 
means  deserve  special  mention  which  have  special  mathematical  or 
empirical  interest."  Messedaglia  also  remarks  ( "  Calcul  des  valeurs 
moyennes,"  Annales  de  demographic  internationale,  1880,  p.  388) 
that  one  can  conceive  of  an  unlimited  number  of  means  of  various 
kinds.  Messedaglia  mentions  that  the  Roman  philosopher  and 
mathematician,  Boetius  (470-525  A.D.),  enumerated  ten  means  in 
bis  work,  De  Arithmetica. 

131 


132  THE  VARIOUS  KINDS  OF  AVERAGES 

Aside  from  the  means  named  there  are  others  which 
originate  from  the  items  of  the  series  through  the  assump- 
tion of  other  mathematical  principles,  for  example,  the 
harmonic  mean,^  the  contraharmonic  mean,^  and  the  mean 
square  or  quadratic  mean.*     In  addition,   Fechner  men- 

'  Messedaglia  developed  especially  the  harmonic  mean  and  its  rela- 
tions to  the  arithmetic  and  geometric  means.  ("Calcul  des  valeurs 
moyennes,"  Annales  de  deraographie  Internationale,  1880.)  The  for- 
mula for  the  harmonic  mean   (M)   between  the  two  values,  a  and  b, 

is       ^ — .     This  formula  does  not  hold  for  more  than  two  values. 

a  +  b 
For  the  case  of  several  values,  ai,  as,  as  .  .  .  .  &nf  Messedaglia  gave 
several  formulas  of  which  the  following  is  the  best  known: 

M  = B 


111  1 

+ H + +  

ai        Hi        as  an 

(Cf.  Messedaglia,  p.  397,  and  Blaschke,  Vorlesungen  iiber  mathe- 
matische  Statistik,  p.  71.)  The  harmonic  mean  of  several  values  is 
always  less  than  the  arithmetic  or  geometric  mean  of  these  values. 
Thus  the  values  1  and  2  have  the  harmonic  mean  1.33,  the  geo- 
metric mean  1.41,  and  the  arithmetic  mean  1.50.  Of  the  three 
means  of  the  same  set  of  values  the  geometric  is  always  the  geo- 
metric mean  of  the  other  two.  Given  any  two  of  the  three  means 
the  third  may  be  found  from  them.     (Messedaglia,  v.  390.) 

*  The  formula  for  the  contraharmonic  mean  is 

, J  _  a.^  4-  ajjg  + +  an^ 

~  ai   +  ai  + +  an  * 

The  contraharmonic  mean  of  several  values  is  always  greater  than  the 
harmonic,  the  arithmetic,  and  the  geometric  means.  The  contrahar- 
monic mean  of  1  and  2  is  1.G6.  The  arithmetic  mean  of  any  set  of 
values  always  equals  the  arithmetic  mean  of  the  harmonic  and  contra- 
harmonic means  of  the  same  set.  Any  one  of  these  means  can, 
therefore,  be  computed  from  the  other  two.     (Messedaglia,  p.  393  f. ) 

*  The  quadratic  mean  of  the  values     ai,  aa,  as    ....  an     is  the 
square   root  of   the   arithmetic   mean  of   the   squares   of   the  values 

der  kleinen  Zahlen,  p.  9.)  This  mean  is  used  by  Gauss  in  the 
theory  of  error  in  computing  the  mean  error.     The  quadratic  mean 


ai'^  +  3.2- +  a 


-  ).       (Cf.  von  Bortkiewicz,  Das  Gesetz 


SYNOPSIS  133 

tions  the  ' '  Scheidewert, ' '  the  ' '  schwersten  Wert, ' '  and  the 
* '  Abweichungsschwerwert. ' '  ^  Although  none  of  these 
means,  except  the  standard  deviation,  are  actually  used  to 
characterize  statistical  series,  yet  some,  for  instance  the  har- 
monic and  contraharmonie  means,  have  been  investigated 
and  discussed  because  of  their  interesting  mathematical 
properties. 

Of  the  means  used  in  statistics  the  arithmetic  average 
is  the  most  important.  Mathematicians  and  statisticians 
have  always  used  it  and  investigated  its  properties.  The 
geometric  mean  is,  likewise,  familiar  to  mathematicians  but 
seldom  used  in  statistics.^* 

The  median  and  mode  have  been  developed  recently, 
mainly  by  Fechner  and  the  English  statisticians  and  widely 
applied  by  them.  Yet,  Messedaglia,  undoubtedly  one  of  the 
foremost  statisticians  of  his  time,  in  his  paper  entitled 
''  Calciil  des  valeurs  moyennes,"  published  in  the  Annates 
de  demograpMe  Internationale  in  1880,  did  not  mention 
these  at  all.  The  three  "  classical  "  means  which  were  his 
principal  objects  of  investigation  were  the  arithmetic,  ge- 
ometric, and  harmonic  means,  all  of  which,  Messedaglia 
stated,  were  known  to  Plato  and  Aristotle  and  considered 
by  Boetius,  the  Roman  mathematician  and  philosopher,  in 
his  De  arithmetica. 


of  several  values  is  always  greater  than  the  arithmetic  mean  and 
less  than  the  contraharmonie  mean;  it  is  identical  with  the  geometric 
mean  of  the  two  last  named  means.      (Messedaglia,  pp.  394,  402.) 

"  See  Fechner,  Kollektivmasslehre,  pp.  160,   172-181. 

°a  The  geometric  mean  was  applied  to  price  statistics  by  W.  Stan- 
ley Jevons  in  his  well-kno\vn  study,  A  Serious  Fall  in  the  Value  of 
Gold  Ascertained  and  Its  Social  Effects  Set  Forth,  published  in  1863 
(reprinted  by  the  Macmillan  Co.  in  1884  in  Investigations  in  Cur- 
rency and  Finance).  F.  Y.  Edgeworth  (Jour.  Roy.  Stat.  Soc,  Vol. 
XLVI,  p.  714),  Francis  Galton  (Proc.  Roy.  Soc,  Vol.  XXIX,  p.  365), 
Donald  McAlister  (Proc.  Roy.  Soc,  Vol.  XXIX,  p.  367),  and  A.  W. 
Flux  (Quar.  Jour.  Econs.,  Vol.  XXI.  p.  613)  have  discussed  the  use 
of  the  geometric  mean  in  statistics. — Teanslatob. 


134  THE  VARIOUS  KINDS  OF  AVERAGES 

A  statistical  average  falls  into  one  of  two  groups  accord- 
ing as  all,  or  only  a  portion,  of  the  items  of  the  series  are 
required  in  its  computation.  On  the  one  hand,  the  com- 
putation of  arithmetic  and  geometric  means  depends  upon 
all  the  items  of  the  series.  On  the  other  hand,  the  mode 
and  median  are  single  definite  items  chosen  to  characterize 
the  series  because  of  their  positions  in  it. 

Fechner  has  offered  another  classification  of  means  which 
will  be  given  although  it  is  of  more  significance  to  mathe- 
matics than  to  statistics.  His  groups  are,  first,  power 
means,  second,  means  "  which  contest  with  power  means 
for  their  name,"  and  third,  combination  means.  A  power 
mean  is  a  value,  the  absolute  sum  ^^  of  like  powers  of 
the  deviations  of  the  items  from  which  is  a  minimum.^ 
The  best  known  power  means  are  the  median  and  the  arith- 
metic average  for  which  the  absolute  sum  of  the  first  powers 
and  squares,  respectively,  of  the  deviations  of  the  items 
are  minima.     The  second  group  of  means  are  those  of  the 

form   M;n=i/— .      In  this  group  the  arithmetic  mean 

is  of  the  first  order,  and  the  mean  square  of  the  second 
order.  In  the  third  group  of  "  combination  means  "  the 
arithmetic  mean  is  likewise  of  the  first  order,  the  geometric 
mean  being  of  the  fourth  order.^  According  to  Fechner 
it  is  to  be  noted  that  ''  the  arithmetic  mean  is  common  to 

"b  "  Absolute  sum  "  means  that  all  deviations  are  to  be  considered 
positive. — Translatob. 

•  "  Uber  den  Ausgangswert  der  kleinsten  Abweichungssumme,"  Ab- 
handlungen  der  konigl.  sachs.  Gesellschaft  der  Wissenschaften,  Vol. 
XVIII,  p.  37  f. 

^  Ibid.  p.  74.  The  notation  in  the  formula  has  the  following  sig- 
nificance : 

n  r=  power 

2=:"  sum  of  such  terms  as " 
a  zz:  item  of  the  series 

m  =z  number  of  items  — Tbanslatob. 

"  Ibid.  p.  75  f. 


SYNOPSIS  135 

these  three  groups  defined  by  distinct  principles  and  there- 
fore, if  we  do  not  have  special  reasons  for  using  another 
mean,  this  fact  gives  us  a  reason  for  choosing  it  as  the 
most  satisfactory  mean." 

Fechner  also  has  undertaken  to  obtain  averages  (which 
he  calls  main  values)  on  the  basis  of  logarithmic  treatment 
of  series  of  single  observations.^  From  the  logarithms  of 
the  various  members  of  a  series  he  computed  averages  (so- 
called  logarithmic  main  values)  in  order  to  investigate  in 
this  way  "  the  logarithmic  deviations  "  of  the  items,  i.e., 
the  deviations  of  the  logarithms  of  the  items  from  the 
"  logarithmic  main  values."  The  "  logarithmic  main 
values  "  which  he  computed,  were  first,  the  mode  of  the 
logarithms  of  the  items  (which  must  not  be  mistaken  for  the 
logarithm  of  the  mode  computed  from  the  items  themselves), 
second,  the  median  of  these  logarithms  (the  "  logarithmic 
median  "),  and  third,  their  arithmetic  mean.  From  the 
logarithmic  main  values  Fechner  secured  the  natural  values 
corresponding  to  these  logarithms  from  the  logarithm  table. 
He  called  the  natural  or  numerical  value  of  the  logarithmic 
mode  "  the  proportional  mode  "  because  it  is  the  character- 
istic of  this  value  that  in  equal  proportional  distance  from 
it  in  both  directions  more  values  are  united  than  in  the 
same  proportional  distance  from  any  other  value.  This 
"  proportional  mode  "  differs  from  the  arithmetic  mode. 
Fechner  discovered  further  that  the  numerical  value  be- 
longing to  the  logarithmic  median  coincides  with  the  median 
obtained  directly  from  the  items  themselves.  The  natural 
value  of  the  arithmetic  mean  of  the  logarithms  of  the  items 
is  identical  with  the  geometric  mean  of  the  items. 

Unfortunately  the  terminology  in  the  field  of  averages 
is,  as  yet,  uncertain.     In  German  the  words  "  Mittelwerte  " 


•  Cf.  Kollektivmasslehre,  pp.  24  f.,  79-83,  339-351 ;  see  also  "  Uber 
den  Ausgangswert  der  kleinsten  Abweichungssumme,"  p.  14  f. 


136  THE  VARIOUS  KINDS  OF  AVERAGES 

and  "  Durchschnittswert  "  may  each  indicate  merely  the 
arithmetic  average,  or  averages  in  general.  In  English 
the  existence  of  two  expressions,  i.  e.,  "  average  "  and 
* '  mean, ' '  has  led  to  attempts  to  make  a  distinction  between 
these  two  expressions/*'  Bowley  thinks  that  it  would  be  best 
to  use  "  average  "  for  a  purely  arithmetic  concept,  such  as 
the  average  duration  of  life  of  a  mixed  population.  This 
average  duration  of  life  does  not  hold  for  any  constituent 
homogeneous  group  of  the  population  and  is  only  a  short 
expression  for  the  result  of  a  certain  arithmetic  operation; 
the  word  ''  mean,"  however,  ought  according  to  Bowley 
to  be  used  for  objective  quantities  such  as  the  mean  height 
of  the  English  people  around  which  mean  all  the  different 
measurements  group  themselves  with  definite  regularity. 
Bowley 's  proposition  apparently  is  a  result  of  the  distinc- 
tion between  "  at>i)ical,"  and  "  typical  "  means;  the 
former  would  be  "  averages,"  the  latter  "  means."  The 
consequence  of  Bowley 's  terminology,  however,  is  that  there 
would  be  no  English  word  left  for  the  general  idea  of 
the  mean.  In  fact  the  English  idiom  is  very  uncertain. 
The  expressions  ''average  "  and  "  mean  "  are  used  generi- 
cally  as  well  as  to  indicate  the  arithmetic  mean  in  par- 
ticular,^^   In    this    treatise    the    words    "  average  "    and 


^"Elements  of  Statistics,  2nd  ed.,  p.  107, 

"  Thus,  Venn  in  his  paper  "  On  the  Nature  and  Uses  of  Averages  " 
(Jour,  of  the  Royal  Stat.  Soe.,  1891,  p.  430)  uses  the  word  "mean" 
for  the  arithmetic  mean;  on  the  other  hand  the  word  "average"  is 
very  often  employed  for  this  mean.  Thus  in  the  special  report  of 
the  United  States  Bureau  of  the  Census  on  Employees  and  Wages 
(1903,  p.  xxvii),  it  is  used  in  contrast  to  the  median.  The  theoretical 
English  statisticians  frequently  use  the  words  "  average "  and 
"  mean  "  in  a  generic  sense  and  they  select  more  specific  terms  for 
the  different  types  of  means. 

Likewise  the  Italian  terminology  appears  to  vary.  For  instance, 
Colajanni,  to  be  sure  in  opposition  to  the  great  majority  of  Italian 
statisticians,  includes  only  the  arithmetic  and  geometric  means  under 
the  term  "  Valori   medJ,"   and  calls   the  median  and  mode   "  other 


SYNOPSIS  137 

"  mean  "  are  both  used  to  denote  the  general  concept 
unless  the  context  clearly  indicates  reference  to  some  par- 
ticular mean. 

values"  which  may  be  used  to  characterize  series  (Manuale  di 
Statistica  teorica,  p.  181). 

The  French  statisticians  usually  employ  "  moyenne  "  for  average 
in  the  broader  sense  and  more  specific  terms  for  the  various  kinds 
of  averages  (moyenne  arithm^tique,  geometrique,  etc.)  ;  yet  the  arith- 
metic mean  is  many  times  simply  designated  as  "  moyenne." 


CHAPTER  II 

THE  ARITHMETIC  MEAN 

A.    THE  SIMPLE  ARITHMETIC  MEAN,  OR,  SHORTLY, 
ARITHMETIC  MEAN 

1.    CONCEPT  AND  QUALITIES  OP  THE  ARITHMETIC  MEAN 

The  simple  arithmetic  mean,  the  most  widely  known 
and  used  statistical  mean,  is  computed  by  dividing  the 
sum  of  the  items  by  their  number.  The  arithmetic  mean 
denotes  the  size  which  the  items  would  have  if,  the  sum 
total  remaining  unchanged,  they  would  all  be  made  equally 
large.  The  statement  of  this  value  carries  with  it  im- 
portant information  about  the  series  from  which  the  arith- 
metic mean  was  computed. 

From  the  manner  of  computing  the  arithmetic  mean  it 
follows  directly  that  the  sum  of  the  positive  deviations 
from  the  arithmetic  mean  is  equal  to  the  sum  of  the 
negative  deviations  from  that  mean.  According  to  another 
mathematical  theorem  the  arithmetic  mean  of  a  series  of 
items  is  characterized  by  the  fact  that  the  sum  of  the 
squares  of  the  deviations  of  the  items  from  the  mean  is  a 
minimum. 

From  the  definition  of  the  arithmetic  mean  it  follows 
that  its  value  will  be  affected  by  a  change  in  any  member 
of  the  series.  This  is  not  the  case  with  other  means.  The 
median  and  the  mode,  for  instance,  may  remain  unchanged 
even  if  considerable  parts  of  the  series  are  changed,  since 
these  means  are  not  computed  from  all  the  items  but  are 
found  by  choosing  one  item  to  represent  the  series  because 
of  the  characteristic  position  of  that  item  in  the  series. 

138 


THE  ARITHMETIC  MEAN  139 

Therefore,  median  and  mode  can  only  be  found  in  series 
the  members  of  which  are  arranged  according  to  magnitude. 
The  arithmetic  mean,  however,  does  not  presuppose  any 
definite  arrangement  of  the  items  and  the  same  value  is 
obtained  by  adding  the  items  in  any  order. 

Furthermore,  the  arithmetic  mean  has  this  advantage 
over  the  median  and  the  mode  that  it  can  be  computed 
from  every  series  of  items  while  the  latter  can  only  be 
obtained  in  series  of  individual  observations,  and  even  in 
these  cases  the  mode  cannot  be  computed  unless  there 
be  a  decided  point  of  concentration. 

2,   DISTINCTION  BETWEEN  STATISTICAL  SERIES  WITH  REFERENCE 
TO  THE  COMPUTATION  OF  THE  ARITHMETIC  MEAN 

The  consideration  of  the  various  conditions  under  which 
arithmetic  means  are  computed,  leads  us  back  to  our  divi- 
sion of  statistical  series  into  three  groups.  These  were,  first, 
series  of  individual  observations;  second,  series  the  mem- 
bers of  which  indicate  the  size  of  quantities  that  are  lim- 
ited in  a  certain  way  (constituents  of  a  totality)  ;  third, 
series  the  members  of  which  characterize  definitely  lim- 
ited quantities  (parts  of  a  larger  whole)  in  a  certain  man- 
ner by  relative  numbers  or  means. 

From  the  series  of  the  first  two  groups  arithmetic  means 
can  be  computed  directly  by  dividing  the  sum  of  the 
items  by  their  number.  The  mean  can  be  computed 
directly  from  these  series  even  if  they  consist  of  sub- 
ordinate numbers.  In  a  series  of  the  first  group  subordi- 
nate numbers  indicate  what  percent  of  the  single  cases 
fall  into  the  various  numerical  classes.  Then  the  average 
magnitude  of  the  element  under  observation  is  computed 
by  treating  the  subordinate  numbers  like  absolute  numbers 
and  by  dividing  the  sum  total  of  the  series  by  100.  If 
wage  data  are  under  consideration  and  if  20^  of  the 
workmen    whose  wages  were   ascertained    receive   $20.00 


140  THE  VARIOUS  KINDS  OF  AVERAGES 

per  week,  30^  receive  $22.00,  20^  receive  $24.00,  20^  receive 
$26.00  and  10^  receive  $28.00,  then,  in  order  to  obtain  the 
average  wage,  the  various  wage  items  are  multiplied  with 
the  corresponding  percentage  and  the  sum  is  then  divided 
by  100.  In  this  way  it  will  be  found  that  the  average  wage 
is  $23.40.1- 

If  a  series  of  the  second  group  consists  of  subordinate 
numbers,  they  indicate  what  percent  of  the  totality  of  a 
higher  order  falls  to  the  constituents.  In  order  to  find 
the  percentage  of  the  totality  which  on  the  average  falls 
to  a  constituent — this  percentage  depending  merely  on 
the  number  of  the  constituents — it  is  merely  necessary  to 
divide  100  by  the  number  of  constituents.  If  we  have  a 
series  of  subordinate  numbers  which  indicate  what  per 
cent  of  all  the  deaths  of  a  year  occur  in  each  month,  then 
the  average  percentage  per  month  (needed  to  ascertain 
what  months  are  above  and  what  are  below  the  average) 
is  found  by  dividing  100  by  12,  giving  8.3^. 

We  now  come  to  series  of  the  third  group,  the  members 
of  which  are  relative  (subordinate  or  coordinate)  numbers, 
or  averages.  However,  we  shall  not  here  consider  these 
series  if  their  members  are  other  than  relative  members 
or  arithmetic  means  (for  instance,  medians  or  modes)  since 
only  higher  means  of  the  same  kind  (i.  e.,  also  medians 
or  modes)  may  be  contrasted  to  these  items,  while  the 
computation  of  an  arithmetic  mean  from  the  items  is  ex- 
cluded.^^  But  in  no  case  should  a  simple  arithmetic  average 
be  computed  directly  from  the  items  of  a  series  of  the  third 
group.  The  members  of  such  series  as  a  rule  refer  to 
different  quantities  (constituents)  and,  consequently,  are 
of  different  weight,  while  the  relative  importance  of  the 
different  members  is  not  clear  from  the  series  itself.  If 
we  have  a  series  of  death  rates  for  different  years,  terri- 

"  (20  X  $20)   +   (30  X  $22)   -f  (20  X  $24)   +   (20  X  $26)  + 
( 10  X  $28 )  =  $2,340  =  100  X  $23.40. 
^'P.  17  f.  and  p.  22  f. 


THE  ARITHMETIC  MEAN  141 

tories,  professions  or  ages,  then  different  "  weights  "  must 
be  given  to  these  numbers  because  they  correspond  to  frac- 
tions with  different  denominators  on  account  of  the  change 
in  the  population  in  the  course  of  years,  or  the  various 
population  of  the  territories,  or  various  numbers  in  the 
professions  or  ages.  If  we  treat  the  single  members  of 
such  a  series  as  equal  and  compute  a  simple  arithmetic 
mean  directly  we  arrive  at  a  wrong  result.  In  such  a  case 
the  mean  must  under  no  condition  be  computed  directly 
from  the  items,  but  independently,  on  the  basis  of  the 
corresponding  data  for  the  entire  quantity  in  question. 
Thus  with  reference  to  the  death  rates  for  the  various 
parts  of  a  country  or  groups  of  population  the  mean  death 
rate  for  the  entire  population  must  be  computed  by  bring- 
ing the  number  of  the  whole  population  into  independent 
relation  with  all  the  deaths  having  occurred.  The  average 
death  rate  per  year  is  obtained  by  dividing  the  total  number 
of  deaths  for  the  period  in  question  by  the  sum  total  of 
the  yearly  population,  or  by  dividing  the  average  death 
rate  by  the  average  population  for  the  period.  The  value 
thus  computed  is  the  weighted  arithmetic  mean  of  the 
members  of  the  series,  that  is,  the  death  rate  for  the  whole 
population  is  the  weighted  arithmetic  mean  of  the  death 
rates  for  the  single  provinces,  or  the  different  groups  of 
population  to  which  the  items  refer.  The  constituents  have 
contributed  to  the  resulting  average  according  to  their 
weights.  Thus  the  general  average  wage  of  the  workmen 
of  a  certain  territory  forms  the  weighted  arithmetic  mean 
of  the  average  wages  for  certain  categories  of  workmen, 
and  the  general  average  duration  of  life  forms  the  weighted 
arithmetic  mean  of  the  values  for  the  average  mean  dura- 
tion of  life  of  the  people  belonging  to  different  groups  of 
the  population.  Consequently,  if  we  desire  to  compute 
the  true  arithmetic  mean  for  a  totality  from  its  con- 
stituents (not  having  data  complete  enough  to  use  the 
method  described  above)   it  is  necessary  to  estimate  the 


142  THE  VARIOUS  KINDS  OF  AVERAGES 

relative    importance    of   the   constituents   and   find   their 
weighted  average. 

Although  it  is  the  rule  that  relative  numbers  or 
averages  which  refer  to  differences  of  time,  space,  and 
quality  or  quantity  (constituents  of  a  larger  totality)  are 
of  unequal  weight,  yet  cases  may  occur  where  these  in- 
equalities are  so  trifling  that  they  need  not  be  taken  into 
account.  Especially  in  time  series  in  the  field  of  popula- 
tion statistics  the  items  very  often  differ  only  slightly  in 
weight,  if  the  population  has  not  changed  considerably  in 
the  course  of  the  years  under  consideration.  In  such  cases 
a  simple  arithmetic  mean  may  be  computed  directly  from 
the  items,  if  necessary,  without  resulting  in  a  large  error. 
If  the  items  are  of  entirely  equal  weight,  then  the  value 
computed  independently  for  the  totality  is  the  simple 
arithmetic  mean  of  the  items  and  is  identical  with  the 
value  which  is  obtained  directly  from  the  items.  In  such 
cases  the  method  of  computation  is  merely  a  question  of 
convenience,  dependent  upon  the  material  at  hand. 

3.    COMPUTATION  OP  THE  ARITHMETIC  MEAN 

The  computation  of  the  arithmetic  mean,  from  its  known 
items,  is  purely  mechanical,  the  simple  arithmetic  opera- 
tions required  being  Imown  to  everybody.  However,  the 
statistician  is  quite  frequently  confronted  by  the  task  of 
computing  averages  from  series  ^*  that  do  not  exhibit  the 
original  items  individually  but  that  consist  of  classes,  i.  e., 
the  series  merely  indicating  how  many  items  there  are 
between  certain  limits.  It  cannot  be  seen  from  such  series 
in  what  manner  the  items  belonging  to  the  single  classes 
are  distributed  between  their  limits.  The  computation  of 
the  arithmetic  mean,  however,  presupposes,  at  least  theo- 
retically, the  knowledge  of  all  the  single  members  of  a 
series,  since  they  must  be  added.     In  order  to  be  able  to 

^*  Cf.  p.  89  f. 


THE  ARITHMETIC  MEAN  143 

compute  arithmetic  means  from  series  that  consist  of  classes 
we  usually  resort  to  hypotheses  as  to  the  grouping  of  the 
items  within  the  classes,  and  then  we  use  the  values  cor- 
responding to  that  particular  hypothesis  in  the  computa- 
tion. The  hypothesis  that  the  items  are  distributed  uni- 
formly in  the  single  classes  is  most  frequently  used  in  this 
connection.  Therefore  the  mid-value  of  each  class  is  taken 
as  the  average  of  all  the  items  belonging  to  that  class,  and 
is  used  in  the  computation  of  the  arithmetic  mean  for 
the  whole  series.^^ 

The  actual  grouping  of  the  items  within  the  different 
classes,  of  course,  never  agrees  completely  with  the  hypoth- 
esis of  uniform  distribution.  If  classes  of  wide  limits 
are  given,  the  hypothesis  of  uniform  distribution  of  the 
items,  in  most  cases,  is  incorrect.  An  example  of  the 
difficulties  that  must  be  overcome  in  such  a  series  is  given 
in  the  computation  of  the  average  age  of  marriage  from 
the  data  as  published  by  most  statistical  bureaus."  In 
these  publications  the  ages  of  those  marrying  are  usually 
presented  in  classes  of  several  years  each.  Since  the  fre- 
quency of  marriage  changes  considerably  with  age,  it  is 
clear  that  those  marrying  cannot  be  distributed  uniformly 
within  the  given  age  classes.  If  classes  of  ten  years  each 
are  formed  the  hypothesis  of  the  uniform  distribution 
cannot  be  used  even  for  one  single  class  without  resulting 
in  a  considerable  error.  In  the  class  that  contains  the 
ages  20  and  less  than  30  years,  the  males  will  probably 
be  more  densely  crowded  together  in  the  years  at  the  end 

*°  Now  and  then  other  hypotheses  are  used  in  the  computation  of 
the  arithmetic  mean  of  a  series  consisting  of  classes.  Thus  in  the 
special  report  of  the  U.  S.  Bureau  of  the  Census  Employees  and 
Wages  (1903,  p.  xxvii),  arithmetic  means  are  found  in  the  com- 
putation of  which  "  the  lowest  wage  in  each  wage  group  was  taken 
as  the  exact  wage  for  each  individual  in  the  group  "  ( ibid,  note  1 ) . 
This  is  a  simplified  procedure,  but  theoretically  not  quite  correct. 

''  Compare  with  this  the  remark  of  G.  v.  Mayr  in  Bevolkerungs* 
statistik,  p.  402. 


144  THE  VARIOUS  KINDS  OF  AVERAGES 

of  the  period  than  in  the  years  at  the  beginning,  while 
the  majority  of  the  females  will  probably  belong  in  the 
first  half  of  this  period.  In  the  class  30  and  less  than 
40  years  of  age,  both  sexes  undoubtedly  will  be  more 
densely  crowded  in  the  first  part  and  the  farther  we 
progress  the  rarer  they  will  be.  Therefore,  it  is  incorrect 
to  assume  that  all  those  in  the  class  20  and  less  than  30 
years  are  25  years  old  on  the  average  and  all  those  30 
and  less  than  40  years,  35  years  old  on  the  average.  In 
fact  the  average  age  of  males  of  the  first  class  (20  and  less 
than  30  years)  is  higher,  and  that  of  females  of  this  class 
as  well  as  the  average  age  of  both  sexes  in  the  second  class 
(30  and  less  than  40  years)  is  lower  than  results  from  the 
hypothesis  of  uniform  distribution.  In  order  to  compute 
the  average  age  of  all  those  marrying  we  must  obtain,  first 
of  all,  the  average  age  in  each  class.  But  how  may  these 
class  averages  be  estimated  without  the  aid  of  more  de- 
tailed data? 

Theoretically,  of  course,  the  difficulty  in  computing  the 
average  of  a  series  which  consists  of  classes,  is  always  the 
same,  no  matter  if  these  classes  are  narrow  or  wide.  But 
the  errors  that  may  result  if  the  hypothesis  of  uniform 
distribution  is  used  for  wide  classes  are  far  greater  than 
for  narrow  classes,  for  instance,  age  classes  of  one  year. 
The  age  distribution  of  the  living  is  usually  given  in  one- 
year  classes.  But  even  here  the  hypothesis  of  uniform 
distribution  in  the  classes  is  not  always  free  from  objec- 
tions. In  the  higher  age  classes  the  distribution  within 
the  single  year  is  certainly  not  uniform  but  decreases 
towards  the  end  and,  consequently,  the  hypothesis  of  uni- 
form distribution  would  result  in  an  average  age  which 
is  somewhat  too  high.  However,  this  error  will  be  com- 
paratively small.^^  Therefore  it  often  serves  the  purpose 
to  first  divide  larger  classes  by  interpolation  into  smaller 

*^  Compare  with  this  G.  v.  Mayr,  Bevolkerungsstatistik,  p.  84. 


THE  ARITHMETIC  MEAN  145 

classes  and  then  to  compute  the  arithmetic  mean  from  the 
latter  by  using  the  hypothesis  of  uniform  distribution. 

Among  the  series  consisting  of  classes  there  are  such 
whose  first  and  last  classes  are  limited  only  in  one  direction. 
With  such  series  the  computation  of  the  average  is  espe- 
cially difficult.  The  following  represents  such  a  series  for 
the  ages  of  those  marrying :  under  20  years,  20  and  less  than 
25  years,  25  and  less  than  30  years,  30  and  less  than  40 
years,  40  and  less  than  50  years,  50  years  and  more.  The 
lower  limit  of  the  first  class  and  the  upper  limit  of  the 
last  class  are  unknown.  However,  the  items  which  belong 
to  the  first  class  (under  20  years)  cannot  go  below  the  legal 
age  of  marriage,  while  the  items  of  the  class  ' '  50  years  and 
more  "  are  limited  by  the  maximum  duration  of  life.  But 
these  are  extreme  limits  and  the  items  undoubtedly  do  not 
extend  quite  so  far  in  reality.  The  items  "  under  20 
years  "  will  all  be  close  to  20  years  and  the  items  "  50 
years  and  more  "  close  to  50  years.  Other  details  are  not 
known.  Therefore  it  is  necessary  to  estimate  rather  arbi- 
trarily the  average  age  of  those  marrying  "  under  20 
years  "  and  "  50  years  and  more  "  as  a  preliminary  to 
the  computation  of  the  average  age  of  all  those  marrying. 

An  interesting  example  of  the  computation  of  an  average 
from  a  series  consisting  of  classes  that  has  no  maximum 
limit,  is  given  in  the  treatment  of  the  statistics  of  tourists 
in  the  publication  of  the  Austrian  Treasury  Department 
Daten  zur  Zalilungshilanz}^  The  length  of  the  sojourn 
of  tourists  in  certain  places  is  registered  as  follows:  up 
to  3  days,  from  3  to  7  days,  from  1  to  2  weeks,  from  2  to 
3  weeks,  from  3  to  4  weeks,  from  4  to  5  weeks,  from  5  to  6 
weeks,  longer  than  6  weeks.  In  the  publication  quoted  the 
hypothesis  of  uniform  distribution  is  used  for  all  classes 
with  the  exception  of  the  lowest  (up  to  3  days)  and  of  the 
last  class  without  maximum  limit  (longer  than  6  weeks), 

"  Tabellen  zur  Wahrimgsstatistik,  2n(i  ed.,  Pt.  II,  No.  3,  p.  829. 


146  THE  VARIOUS  KINDS  OF  AVERAGES 

i,  e.,  in  all  classes,  with  the  exception  of  the  two  named,  it  is 
assumed  that  the  average  length  of  the  sojourn  may  be 
expressed  by  the  arithlnetic  mean  of  the  two  limits  of  the 
classes  in  question.  In  the  class  "up  to  3  days,"  which, 
under  the  above  assumption,  would  give  the  arithmetic 
mean  of  ''2  days,"  an  average  length  of  sojourn  of  1.2 
days  is  assumed.  The  average  sojourn  of  people  who 
at  registration  were  put  in  the  class  ' '  longer  than  6  weeks  ' ' 
is  supposed  to  be  50  days.  On  the  basis  of  these  averages 
assumed  for  the  single  classes  (and  under  the  assumption 
of  an  average  sojourn  of  2  days  for  those  persons  about 
whose  length  of  stay  no  declaration  was  made)  the  total 
average  for  the  entire  series,  i.  e.,  the  average  length  of 
sojourn  of  all  tourists  together,  is  found  to  be  8.5  days.^^ 

Complete  statistical  series,  that  is  those  whose  items  are 
given  in  detail,  as  well  as  series  consisting  of  classes,  are 
sometimes  subjected  to  adjustment,  in  order  to  remove 
the  more  or  less  accidental  unevenness  in  the  formation 
of  the  series  or  in  the  form  of  the  curve  resulting  from 
graphic  representation  of  the  series.  For  statistical  series 
usually  exhibit  irregularities  in  the  details,  even  if  a  cer- 
tain characteristic  formation  can  be  recognized.  These 
can  be  traced  back  to  the  inevitable  accidental  errors  which 
every  empirical  determination  of  a  value  shows,  to  the  lim- 
itation of  the  field  of  observation  and  to  the  imperfections 
of  the  observation  (for  instance,  incorrect  declaration  of 
age).  In  addition  to  these  there  jaay  occur  special  dis- 
turbances in  the  normal  course  of  the  observed  phenomena 
(for  instance,  epidemics  in  the  case  of  mortality  statistics).^" 

*•  Fechner  has  developed  a  special  mathematical  procedure  for  the 
computation  of  the  arithmetic  mean  of  a  series  without  superior 
and  inferior  limits  ( Kollektivmasslehre,  §128  ["  Supplementarver- 
fahren  "J  ) .  This  procedure,  however,  is  applicable  only  under  the  sup- 
position that  the  series  corresponds  to  the  asymmetrical  Gaussian  law. 

^°  Von  Bortkiewicz  in  the  article  "  Ausgleichung  der  Sterblich- 
keitstafeln "   in  Handw.   d.   Staatsw. ;    compare   also   Czuber,   Wahr- 


THE  ARITHMETIC  MEAN  147 

The  adjustment  may  be  done  graphically  by  construct- 
ing a  curve  which  supposedly  represents  the  essential  char- 
acteristic traits  of  the  structure  of  the  series.  There 
are  also  various  mechanical  methods  of  adjustment  ^^  as 
well  as  those  based  on  mathematical  functions.     The  latter 

methods  are  used  especially  in  the  adjustment  of  mortality 
tables."-22a 

Although  by  adjustment  of  a  series  we  intend  in  the 
first  place  to  improve  its  general  formation,  yet  this  ad- 
justment has  a  special  importance  for  the  determination 
of  the  means  from  the  series  in  question.  By  the  adjust- 
ment the  items  of  the  series  are  modified  and  it  may  hap- 

scheinlichkeitsrechnung,  No.  196,  "  Ausgleichung  von  Tafeln,"  p. 
392  f. 

^'  The  mechanical  methods  of  adjustment  or  graduation  often  de- 
pend upon  computations  of  averages.  Wittstein's  method  is  as 
follows:  He  takes  the  arithmetic  mean  of  each  5  successive  items 
throughout  the  whole  series  to  be  adjusted  and  puts  it  in  the  place 
of  the  middle  item  of  the  group.  Woolhouse  and  Karup  proceed 
by  finding  five  values  for  every  item  of  the  series  to  be  adjusted, 
one  of  these  values  results  from  the  observation  itself  while  the 
other  four  originate  from  interpolation.  The  arithmetic  mean  of 
these  five  values  is  taken  to  be  the  adjusted  value  (cf.  Czuber, 
Wahrscheinlichkeitsrechnung,  p.  403  ff. ). 

^^  Of  the  voluminous  mathematical  literature  on  the  methods  of 
adjustment  may  be  mentioned  especially:  Blaschke,  Die  Methoden 
der  Ausgleichung  von  Massenerscheinungen,  Vienna,  1893,  and  the 
same,  Vorlesungen  iiber  math.  Statistik,  Leipsic,  1906  (particularly 
Pt.  VI)  ;  cf.  also  Czuber,  Die  Wahrscheinlichkeitsreehnimg,  No.  196 
"  Adjustment  of  Tables,"  No.  198  "  Mechanical  Methods  of  Adjust- 
ment "  and  "  Graphical  Adjustment " ;  Bowley,  Elements  of  Statistics, 
2nd  ed.,  pp.  254-258;  Westergaard,  Die  Grundziige  der  Theorie  der 
Statistik,  pp.  130-136,  and  Die  Lehre  von  der  Mortalitat  imd  Mor- 
bilitat,  pp.  Ill  f.  and  202  f. 

''aAlljTi  A.  Young  gives  a  bibliography  on  methods  of  adjusting 
age  data  in  "The  Adjustment  of  Census  Age  Returns"  (Western 
Reserve  Bulletin,  November,  1902).  The  same  writer  also  gives  a 
brief  discussion  and  bibliography  of  this  subject  in  Bulletin  13  of  the 
Bureau  of  the  Census  (pp.  47-53).  Newsholme  describes  easy  graphic 
methods  of  adjustment  in  his  Vital  Statistics. — TRAjfSLATOR. 


148  THE  VARIOUS  KINDS  OF  AVERAGES 

pen  that  the  arithmetic  mean  computed  from  the  adjusted 
series  is  not  exactly  the  same  as  that  which  would  result 
from  the  non-adjusted  series.  Large  differences,  however, 
must  not  be  expected,  since  the  uneven  points  which  were 
removed  by  the  adjustment  probably  would  have  counter- 
balanced each  other  in  the  computation  of  the  arithmetic 
mean.  At  any  rate,  the  size  of  the  arithmetic  mean  of  an 
adjusted  series  may  depend  in  a  certain  measure  on 
the  manner  of  the  adjustment  and  on  the  method  chosen. 
When  computing  the  arithmetic  mean  from  a  series,  as 
mentioned,  the  items  of  the  series  are  added  and  then 
divided  by  the  number  of  members.  Therefore,  in  order 
to  be  able  to  compute  the  arithmetic  mean  from  a  series, 
either  all  the  single  members  of  the  series,  the  sum  of  which 
is  to  be  found,  must  be  known,  or  if  this  is  not  the  case, 
estimated  values  must  be  substituted  for  them.  The 
knowledge  or  the  estimation  of  the  single  members,  how- 
ever, becomes  unnecessary,  if  their  sum  and  number  be 
given.  In  such  a  case  it  is  sufficient  to  divide  the  former 
value  by  the  latter  in  order  to  compute  the  arithmetic  mean. 
To  be  sure,  the  isolated  mean  which  we  find  in  this  way 
gives  only  limited  information  about  the  series.  A  thor- 
ough insight  is  possible  only  when  the  items  are  ascer- 
tained and  arranged  in  a  statistical  series  from  which  we 
may  compute  the  arithmetic  mean,  as  well  as  other  means 
and  the  dispersion  of  the  items  around  the  mean. 

4.   APPLICATION  OF  THE  ARITHMETIC  MEAN 

As  is  well  known,  arithmetic  means  are  used  frequently 
in  all  departments  of  statistics.  First  of  all  the  arithmetic 
means  used  in  the  field  of  population  statistics  must  be 
mentioned:  the  mean  duration  of  life  (of  the  new-born 
or  of  people  at  certain  ages),  the  average  age  of  the  living, 
the  dead,  and  those  marrying,  the  average  number  of  chil- 
dren per  family.    Various  fundamental  questions  concern- 


THE  ARITHMETIC  MEAN  149 

ing  the  manner  of  computation  and  the  usefuhiess  of  these 
averages  have  already  been  touched  upon  in  previous  chap- 
ters. However,  it  would  lead  us  too  far  afield  to  investi- 
gate the  scientific  importance  of  these  averages  and  the 
conclusions  which  can  be  dra\ra  from  them  with  reference 
to  the  peculiarities  of  population  statistics. 

As  an  illustration  of  the  use  of  the  arithmetic  mean  in 
other  fields  we  may  mention :  the  average  height  of  people, 
which  plays  an  important  role  in  anthropological  statistics, 
the  average  temperature,  and  the  average  barometric  height 
in  the  field  of  meteorology,  the  average  number  of  people 
living  per  domicile,  the  average  size  of  landed  properties 
or  of  agricultural  establishments  expressed  by  some  super- 
ficial measure,  the  average  wage,  the  average  income  of 
persons  counted  for  the  income  tax,  the  average  account 
of  a  holder  of  a  savings  bank  account,  the  average  dura- 
tion of  disease,  the  average  distance  covered  by  a  passenger 
or  a  ton  of  freight,  the  average  tonnage  and  the  mean 
cargo  of  a  ship,  the  average  amount  of  a  postal  money 
order,  the  average  number  of  members  of  a  club  or  of  a 
cooperative  society,  the  average  business  share  of  a  member 
of  a  cooperative  society,  and  so  forth. 

The  progressive  development  of  statistical  science  leads 
to  the  continuous  opening  of  new  fields  to  statistical  ob- 
servation ;  new  quantities  are  investigated  statistically  and 
expressed  by  statistical  series.  Every  such  new  series  sug- 
gests the  possibility  of  the  computation  of  an  average. 
At  the  same  time  the  methods  already  in  use  are  refined 
and  new  facts  are  registered.  This  new  information  en- 
ables Tis  to  dissect  the  series  originating  from  the  investiga- 
tions from  new  points  of  view  and  to  divide  them  into 
components  which,  in  turn,  may  give  rise  to  new  averages. 

In  this  connection  we  must  also  mention  the  frequent 
use  of  arithmetic  means  in  graphic  representation.  Since 
the  arithmetic  mean  is  obtained  from  a  series  by  adding 
the  items  and  by  dividing  the  sum  by  the  number  of  the 


150  THE  VARIOUS  KINDS  OF  AVERAGES 

items,  the  siim  of  all  the  items  is  found  by  multiplying 
the  arithmetic  mean  of  a  series  by  the  number  of  its  items. 
This  fact  explains  the  availability  of  the  arithmetic  mean 
for  graphic  representation  in  the  form  of  rectangles.  If 
we  draw  a  rectangle,  its  base  corresponding  to  the  number 
of  items  and  its  height  to  their  average  size,  then  the  area 
corresponds  to  the  sum  of  all  the  items.  Often  different 
quantities  are  represented  graphically  in  this  way  and  then 
their  special  features  are  easily  compared. 

B.     THE  WEIGHTED  ARITHMETIC  MEAN 

The  weighted  arithmetic  mean  (''  das  gewogene  arith- 
metische  Mittel,"  "  moyenne  arithmetique  pesee,"  "  com- 
posee  "  or  '*  graduee,"  "  media  arithmetica  ponderata  " 
or  "  composta  ")  does  not  represent  an  independent  kind 
of  mean.  On  the  contrary  it  agrees  in  its  essential  qualities 
with  the  "  simple  "  arithmetic  mean  and  differs  from  it 
merely  in  one  point  of  secondary  importance,^^ 

This  difference  is  that  in  the  computation  of  a  weighted 
arithmetic  mean  the  items  are  not  simply  added  and  the 
sum  divided  by  the  number  of  items,  but  that  the  items 
before  their  addition  are  multiplied  by  coefficients  (weights) 

"'  The  mean  which  we  call  weighted  arithmetic  mean  here,  was 
formerly  called  geometric  mean  in  Germany,  tv  term  with  which  we 
nowadays  denote  a  kind  of  mean  totally  different  from  the  weighted 
arithmetic  mean,  and  which  will  be  discussed  in  a  later  chapter, 
Haushofer  still  used  the  term  geometric  mean  for  the  mean  which 
in  modern  times  is  called  weighted  arithmetic  mean.  (Lehr-  und 
Handbuch  der  Statistik,  2nd  ed.,  1882,  p.  53.  Such  averages  as  were 
found  with  reference  to  the  relative  weights  of  the  items  of  a  series 
were  called  geometric,  in  opposition  to  the  arithmetic,  which  were 
ascertained  without  reference  to  these  weights.)  Also  G.  v.  Mayr 
called  the  weighted  arithmetic  mean  geometric  mean  in  his  book. 
Die  Gesetzmassigkeit  im  Gesellschaftsleben,  p.  53,  but  in  his  Theo- 
retische  Statistik  of  the  year  1895  he  dropped  this  term  and, 
following  the  English  and  Italian  terminology,  proposed  the  term 
"  weighted  mean,"  which  has  since  been  introduced  generally. 


THE  ARITHMETIC  MEAN  151 

of  different  sizes  and  the  sum  of  the  products  resulting  is 
finally  divided,  not  by  the  number  of  items,  but  by  the 
sum  of  all  the  coefficients.  The  fundamental  principle  of 
the  computation  of  the  two  means  is  the  same.  However, 
when  computing  a  weighted  arithmetic  mean,  the  series  is 
first  subjected  to  a  change  of  formation,  the  purpose  of 
W'hich  is  to  give  to  the  single  members  of  the  series  an 
influence  varying  wdth  their  importance,  their  weights. 

The  example  of  a  weighted  arithmetic  mean  usually 
quoted  is  the  average  price  of  a  commodity  with  reference 
to  the  quantities  sold  at  different  prices.  If  20  units,  yards, 
pounds,  tons,  etc.,  of  a  commodity  are  sold  at  the  price 
of  10,  and  10  units  at  the  price  of  16,  then  a  weighted 
arithmetic  mean  is  computed  from  these  data  by  first 
multiplying  the  prices  by  the  quantities  sold,  adding  these 
products  [(20X10) +  (10X16)  =360]  and  dividing  this 
sum  by  the  number  of  the  units  sold  [30].  In  this  manner 
we  find  the  "  weighted  "  arithmetic  mean,  12.  "Without  ref- 
erence to  the  quantities  sold  we  would  obtain  the  average 
13  from  the  two  prices  10  and  16. 

As  a  matter  of  fact  we  proceed  in  exactly  the  same  way 
if  we  have  wages  and  numbers  of  workmen,  instead  of 
prices  and  quantities  of  merchandise.  When  computing 
the  arithmetic  mean  of  wages  the  procedure  is  self-evident 
and  nobody  thinks  of  speaking  of  a  *  *  weighted  ' '  arithmetic 
mean.  The  series  consists  of  30  independent  units  (work- 
men). In  order  to  simplify  the  series  the  items  of  equal 
value  (equal  wages  of  a  number  of  the  workmen)  are  not 
given  individually.  However,  the  number  of  workmen, 
who  receive  equal  wages  is  known,  and  must  be  taken  into 
consideration.^^* 

"a  Scott  Nearing  defines  the  "simple  mathematical  [arithmetic?] 
average"  erroneously  in  his  Wages  in  the  United  States  (Macmillan, 
1911).  He  says  (p.  120)  that  "the  simple  average,  by  far  the 
least  satisfactory,  is  secured  by  adding  the  rates  of  wages  and  divid- 
ing  by   the    number   of    different   groups   of   wage   ejamers."     This 


152  THE  VARIOUS  KINDS  OF  AVERAGES 

On  the  other  hand  the  price  series,  mentioned  above  as 
an  example,  does  not  consist  of  independently  ascertained 
units.  The  pounds  or  yards,  tons,  etc.,  which  have  been 
sold,  are  merely  arithmetic  units  that  do  not  exist  in- 
dependently. But  different  quantities  of  merchandise  were 
sold  at  different  prices,  and  therefore  the  quoted  prices  have 
different  weights.  Consequently,  it  would  be  insufficient  to 
merely  add  the  different  prices  and  to  divide  their  sum 
by  their  number.  It  is  evident  that  in  the  computation 
of  the  average  the  different  weights  of  the  single  prices 
must  be  expressed.  This  is  done  by  using  as  ''  weights  " 
those  quantities  that  were  sold  at  the  different  prices.  We 
pretend,  so  to  speak,  that  the  series  consists  of  as  many 
members  as  units  of  quantity  sold  and  take  every  quoted 
price  into  account  as  often  as  quantity  units  were  sold  at 
that  price.  Since  this,  however,  is  really  a  pretense,  we 
feel  that  we  deviate  from  the  general  rule  for  the  com- 
putation of  an  arithmetic  mean  and  we  call  the  mean  com- 
puted with  reference  to  the  quantities  sold,  the  "  weighted," 
in  contrast  to  the  simple,  arithmetic  mean. 

When  computing  an  average  price  from  a  time  series  of 
prices  we  ought  to  proceed  in  a  similar  way  as  if  the  several 
prices  were  given  for  the  same  time.  If  in  10  successive 
years  the  quantities  1,  2,  3,  4,  .  .  .  10  are  sold  at  prices 
1,  2,  3,  4,  .  ,  .  10,  then  the  weighted  average-price  for  the 
decade  would  be  7;  i.  e.,  (lXl)+'(2X2)-f(3X3)  +  (4X4) 
+  .  .  .  (10X10)  divided  by  55r:r7.  The  "  simple  " 
arithmetic  mean,  which,  however,  would  be  incorrect,  is  5^/2 
(the  average  of  1,  2,  3,  4,  .    .    .   10). ^^ 

definition  is  at  variance  with  the  definitions  given  by  A.  L.  Bowley 
(Elements  of  Statistics,  p.  109)  and  G.  U.  Yule  (Theory  of  Statis- 
tics, p.  108)  as  well  as  with  the  usage  described  above  by  the  au- 
thor.— Translator. 

"*  The  principle  mentioned  is  taken  account  of  in  the  rules  which 
regulate  the  procedure  of  the  Austrian  permanent  commission  for 
commercial  values  in  the  ascertainment  of  the  annual  average  prices. 


THE  ARITHMETIC  MEAN  153 

In  the  examples  mentioned  the  point  in  question  was  to 
compute  averages  from  items  whose  difference  of  weight 
was  given  numerically.  In  these  cases  the  method  of  com- 
putation of  a  "  weighted  "  arithmetic  mean  is  more  or 
less  self-evident.  Frequently,  however,  it  is  necessary  to 
compute  averages  from  items  which  evidently  have  dif- 
ferent weight,  although  we  have  no  numerical  data  which 
we  could  use  as  "  weights."  If,  in  such  cases,  we  want 
to  express  the  different  importance  of  the  items,  then 
we  are  obliged  to  use  estimated  "  weights."  These 
weights  must  be  chosen  so  that  their  relation  to  each 
other  is  proportionate  to  the  surmised  relation  between  the 
items. 

Cases  of  this  kind  occur  frequently.  Statistical  records 
are  rare  which  state  prices  as  well  as  the  quantities  of 
merchandise  sold.  The  price  lists  of  stock-exchanges  merely 
quote  the  prices  at  which  sales  have  been  made  on  the 
different  days,  but  not  the  quantities  of  stocks  or  merchan- 
dise sold  at  the  different  rates  on  the  different  days.  Con- 
sequently the  computation  of  an  exact  "  weighted  "  aver- 
age for  a  long  period  of  time  is  impossible.  If  quantities 
of  great  difference  were  sold  at  the  different  prices,  we 
could  express  this  fact  when  computing  the  average  for 
a  longer  period  by  the  use  of  estimated  "  weights  "  pro- 
portionate to  the  surmised  quantities  sold.^^ 

There  it  says,  the  commission  must  also  take  into  consideration 
during  what  part  of  the  year  the  greatest  fluctuations  of  price  have 
taken  place  and  how  the  imports  and  exports  of  the  whole  year  are 
distributed  over  the  single  parts  of  the  year,  i.  e.,  the  quantities  im- 
ported and  exported  at  various  times  during  the  year  at  the  different 
prices  must  be  taken  into  consideration  in  connection  with  the  dif- 
ferent price  levels  which  have  existed  at  such  times. 

"'  Average  rates  are  needed  in  order  to  compute  the  revenues  from 
bonds.  Here  annual  average  rates  are  mostly  used  as  bases.  In 
Austria,  if  customs  are  paid  in  silver  (instead  of  in  gold),  a  premium 
must  be  paid,  the  size  of  which  is  determined  monthly  according  to 
the  relation  between  the  monthly  average  rate  of  the  gold  20-frane 


154  THE  VARIOUS  KINDS  OF  AVERAGES 

The  direct  estimation  of  ''  weights  "  is  a  last  resort  to 
be  avoided  if  possible.  If  data  for  the  calculation  of  the 
weights  are  not  at  hand  we  may,  in  order  to  avoid  direct 
estimation,  substitute  numerical  quantities  which  appear  to 
approximate  the  true  weights.  The  following  example  may 
be  given:  In  order  to  compute  accurately  the  average  rate 
of  interest  of  a  bank  for  one  year,  it  would  really  be  nec- 
essary to  combine  the  discount  rates  during  the  year  with 
the  loans  that  have  been  effected  at  such  rates.  This  kind 
of  computation  is  not  practicable.  In  order  to  avoid  a 
direct  estimation  of  the  weights  of  the  different  discount 
rates,  we  take  a  known  measure  which  we  surmise  is  pro- 
portionate to  the  weights  of  the  items,  i.  e.,  to  the  amounts 
of  loans.  In  computing  the  average  rate  of  interest  for  the 
year  we  usually  combine,  therefore,  the  single  discount 
rates  with  the  length  of  time  they  have  ruled,  i.  e.,  we 
multiply  every  discount  rate  by  the  number  of  weeks  it 
was  in  force,  add  these  products,  and  then  divide  the  sum 
by  52,  the  number  of  weeks  in  a  year.  This  assumption  is, 
however,  not  free  from  objection.  For  the  volume  of  loans 
is  not  always  the  same  and  depends  to  a  great  extent  upon 
the  rate  of  interest  itself. 

From  series  of  measurements  (wages,  prices)  the  weights 
belonging  to  the  various  members  of  the  series  can  be  found 
in  the  majority  of  cases.  The  cases  where  this  is  not  pos- 
sible are  usually  occasioned  by  deficient  information  (for 
instance,  if  only  prices  and  not  the  quantities  sold  are 
given).  From  series,  however,  which  consist  of  relative 
numbers  or  means  that  refer  to  constituents  of  a  larger 
totality  (series  of  the  third  group)  the  weights  belonging 
to  the  items  can  never  be  found  directly.  But  we  know 
that  as  a  rule  the  items  of  such  series  have  different  weights. 
If  these  items  are  other  than  arithmetic  means  (medians  or 
modes),  then  it  is  not  appropriate  to  compute  an  arithmetic 

pieces  at  the  Vienna  exchange  and  the  monthly  average  rate  of  the 
coined  silver. 


THE  ARITHMETIC  MEAN  155 

mean  of  the  items.  But  if  the  series  consists  of  relative 
numbers  or  of  arithmetic  means,  then  the  computation  of 
an  arithmetic  mean  from  them  is  undoubtedly  allowable. 
However,  a  weighted,  rather  than  a  simple  arithmetic,  mean 
must  be  found.  Fortunately  the  difficult  computation  of 
such  a  mean  from  the  items  is  usually  not  necessary.  On 
the  contrary,  we  are  frequently  able  to  find  directly  on  the 
basis  of  the  totality  (to  the  constituents  of  \yhich  the  items 
refer)  that  superior  relative  number  or  that  superior  arith- 
metic mean  which  represents  the  weighted  arithmetic  mean 
of  the  items.^**  If  a  series  of  death  rates  is  given  which  re- 
fers to  different  parts  of  the  country  or  to  different  groups 
of  population,  then  the  death  rate  for  the  whole  country 
or  for  the  entire  population  may  be  contrasted  with  these 
items  as  their  weighted  arithmetic  mean.  In  a  similar 
way  the  general  average  wage  of  all  workmen  represents 
the  weighted  arithmetic  mean  for  the  average  wages  of 
certain  categories  of  workmen.  Therefore,  in  series  of  rela- 
tive numbers  and  arithmetic  means  of  the  third  group 
the  computation  of  a  weighted  arithmetic  mean  from  the 
items  is  not  necessary,  if  the  general  relative  number  or 
the  general  arithmetic  mean  is  known  as  ascertainable. 
The  average  should  not  be  computed  from  the  items  but 
rather  from  the  fundamental  data,  on  which  the  items 
themselves  are  based.  The  average  should  be  computed 
from  the  items  themselves  only  when  the  data  necessary 
for  its  independent  computation  are  lacking.  The  numer- 
ical size  of  the  "  weights  "  to  be  used  is  to  be  determined 
in  every  case  with  reference  to  all  circmnstances.  The 
use  of  weights  may  be  dispensed  with  only  in  those  rare 
cases  where  the  items  have  practically  equal  weights.^^ 

The  necessity  for  computing  the  average  directly  from 

items  of  different  weights  occurs  if  estimated  averages  are 

given  as  items — for  instance,  estimated  average  wages  of 

agricultural  laborers  for  the  different  sized  sections  of  a 

*"  P.  140.  *^  P.  142. 


156  THE  VARIOUS  KINDS  OF  AVERAGES 

country.  Since  individual  measurements  are  not  given,  an 
average  for  a  wider  geographical  territory,  the  whole  coun- 
try, cannot  be  computed  on  the  basis  of  the  totality  of 
all  individual  wages.  The  average  for  the  wider  geograph- 
ical territory  can  only  be  found  on  the  basis  of  the  given 
averages  for  the  single  sections.  While  doing  this  we  must 
consider  that  the  number  of  agricultural  laborers  varies 
according  to  the  section  and  that  therefore  the  average 
wages  for  the  single  sections  are  not  of  equal  value.  The 
number  of  laborers  may,  perhaps,  be  obtained  from  the  data 
of  a  census  of  occupations  or  a  census  of  agricultural  estab- 
lishments. If  this  is  the  case,  correct  weights  are  found. 
Otherwise  appropriate  w^eights  must  be  estimated  on  the 
basis  of  other  data. 

The  use  of  weights  in  the  computation  of  mean  index 
numbers  to  represent  changes  in  the  price  level  has  caused 
much  controversy.  As  is  well  known,  the  object  of  this 
computation  is  to  obtain  an  average  from  the  single  index 
numbers  which  denote  the  prices  of  merchandise  of  a  cer- 
tain year  as  a  percent  of  the  prices  of  a  standard  year 
or  period.  This  average  enables  us  to  compare  all  the 
prices  of  any  year  with  the  prices  of  the  standard  year 
or  period,  and  thus  the  prices  of  different  years  with  each 
other. 

The  ordinary  arithmetic  mean  of  the  single  index  num- 
bers seems  to  be  insufficient,  because  in  its  computation 
the  same  importance,  the  same  weight,  is  attributed  to  the 
price  fluctuations  of  all  commodities  under  consideration. 
Thus,  a  fluctuation  in  the  price  of  a  rather  unimportant 
commodity  has  the  same  influence  upon  the  numerical 
value  of  the  mean  index  number  as  a  fluctuation  in  the 
price  of  the  most  important  commodity.  Therefore,  numer- 
ous modem  authors  have  found  it  to  be  necessary  to  com- 
bine the  indices  for  the  single  commodities  with  weights,  in 
order  to  accentuate  the  different  importance  in  commerce 
or  in  consumption.     Some  authors  use  coefficients  chosen 


THE  ARITHMETIC  MEAN  157 

at  will  which,  according  to  their  subjective  judgment,  seem 
to  be  proportionate  to  the  importance  of  the  various  com- 
modities. It  is  equivalent  to  the  use  of  such  coefficients  if, 
in  the  computation  of  the  average,  several  price  quotations 
are  taken  for  one  commodity  or  if  single  commodities  are 
quoted  in  different  stages  of  production,  as,  for  instance, 
Sauerbeck  does.  Instead  of  coefficients  chosen  at  will  we 
may  also  use  coefficients  that  are  computed  on  the  basis 
of  quantities  numerically  known  or,  at  least,  capable  of 
estimation.  Thus  the  Economic  Section  of  the  British  As- 
sociation has  used  the  "  estimated  expenditure  per  annum 
on  each  article  "  as  the  weights  of  the  single  indices.  In 
a  similar  manner  Professor  Conrad,  in  his  works  on  price 
statistics,  assigns  to  the  various  commodities  weights  pro- 
portionate to  their  consumption.  The  British  Board  of 
Trade  obtains  its  mean  index  number  by  computing  the 
value  of  the  foreign  trade  of  a  certain  year,  first,  on  the 
basis  of  the  prices  of  this  year  and  then  on  the  basis  of  the 
prices  of  the  standard  year  by  stating  the  former  value  as  a 
percent  of  the  latter.  The  quantity  of  the  single  commodity 
which  has  been  sold  in  the  foreign  trade  of  that  year  is 
used  as  its  weight.  Vice  versa,  the  value  of  the  trade  of 
the  standard  year  can  be  computed  at  the  prices  of  various 
other  years  and  the  results  can  then  be  compared  with  the 
value  which  the  trade  of  the  standard  year  shows  at  the 
prices  of  such  standard  year.  In  this  computation  the 
quantities  of  the  single  commodities  handled  in  the  foreign 
trade  of  the  standard  year  are  used  as  weights. 

If  mean  index  numbers  are  computed  for  a  series  of 
years,  usually  the  same  weights  are  used  in  the  computa- 
tion of  all  the  total  index  numbers.  Thus  Professor  Con- 
rad, who  considers  the  consumption  of  the  various  com- 
modities to  be  a  measure  of  their  importance,  uses  the 
quantities  consumed  in  the  year  1880  as  fixed  weights 
for  all  former  and  later  years.  But  we  may  also  use 
weights  which  change  in  the  course  of  years  in  the  same 


158  THE  VARIOUS  KINDS  OF  AVERAGES 

manner  as  the  relative  importance  of  the  various  commodi- 
ties.^^ 

Weights  can  also  be  employed  in  a  similar  manner  if  total 
index  numbers  are  used  in  representing  the  change  of  other 
complex  statistical  data.  Thus,  Bowley  has  used  weights  in 
the  computation  of  his  mean  index  numbers  for  the  changes 
of  the  wage  level ;  Wood  used  them  in  computing  indices  to 
represent  the  changes  of  consumption  in  England. 

Bowley  combined  the  indices  which  represent  the  fluctua- 
tion of  wages  in  single  occupations  into  mean  index  num- 
bers for  more  extensive  groups  of  occupations  and  indices 
for  different  localities  into  mean  index  numbers  for  wider 
geographical  territories  and,  in  doing  this,  he  used  weights 
in  order  to  allow  for  the  varying  importance  of  the  occu- 
pations and  localities.^^  He  also  tried,  by  the  use  of  chang- 
ing weights,  to  allow  for  the  changes  which  have  taken 
place  in  the  course  of  time  in  the  different  occupations  and 
localities.^"  Wood  represented  the  quantities  of  various 
articles  of  food  (flour,  cocoa,  coffee,  meat,  rice,  sugar,  tea, 
tobacco,  etc.),  consumed  per  capita  of  the  population  during 
the  years  1860-1896,  by  means  of  index  numbers  as  a  per- 
cent of  the  average  consumption  of  these  articles  during  the 
standard  period  1870-1879,  and  computed  from  these  single 
indices  the  simple  arithmetic  mean,  and  five  kinds  of 
weighted  arithmetic  means  in  order  to  allow  for  the  impor- 

"  Although  Bowley  (Elements  of  Statistics,  Chap.  IX,  "Index 
Numbers,"  p.  220)  does  call  the  prices  of  the  standard  year  weights, 
it  is  an  incorrect  expression.  The  deviation  of  the  prices  of  a  certain 
year  from  the  prices  of  the  standard  year  naturally  depends  on  the 
level  of  the  latter,  therefore  it  is  important  to  choose  a  standard  as 
normal  as  possible.  But  the  height  of  the  prices  of  the  standard  year 
has  no  influence  upon  the  manner  of  the  computation  of  the  mean, 
but  only  upon  the  numerical  size  of  the  items  from  which  the  average 
is  computed  and,  therefore,  on  the  magnitude  of  the  average. 

"  See,  for  instance,  Journ.  of  the  Roy.  Stat.  Soc,  Vol.  LXII 
(1899),  especially  p.  712,  and  Vol.  LXIX   (1906),  p.  164  ff. 

"«  See  Journ.  of  the  Roy.  Stat.  Soc,  Vol.  LXIX  (1906),  p.  167  f. 


THE  ARITHMETIC  MEAN  159 

tance  of  the  different  articles  of  food.  He  computed  the 
weights  used  principally  on  the  basis  of  a  typical  work- 
man's budget  given  by  Booth  in  Life  and  Labor  of  the 
People  of  London.^^ 

IModern  statistics  offers  several  interesting  illustrations 
in  which  a  mean  is  computed  from  a  series  of  items  by 
combining  them  into  a  weighted  mean  index  number,  the 
weights  of  the  items  being  determined  by  the  peculiar  pur- 
pose in  view.  The  United  States  Weather  Bureau  has 
computed  the  average  rainfall,  weighted  according  to  popu- 
lation. The  average  rainfall  of  a  country,  geographically 
considered,  should  be  computed  by  attributing  different 
importance  to  the  different  rainfalls  on  record  according 
to  the  area  covered  by  them.  However,  the  purpose  of  the 
American  statistics  is  to  represent  the  average  rainfall  ac- 
cording to  its  importance  to  the  population.  For  this  pur- 
pose the  various  measurements  of  the  rainfall  are  not 
weighted  according  to  the  areas  covered  by  the  different 
rains,  but  according  to  the  population  of  the  areas.  If  the 
importance  of  the  rainfall  to  the  population  is  to  be  rep- 
resented, then  rainfalls  in  uninhabited  districts  evidently 
need  not  be  taken  into  account.  The  importance  of  rainfall 
varies  with  the  density  of  population.  In  this  sense  the 
average  rainfall  in  the  United  States  has  decreased  from 
42.5  in.,  in  1870,  to  41.4  in.,  in  1890.  But  this  does  not 
prove  that  a  meteorological  or  climatic  change  has  occurred 
but  is  caused  principally  by  the  fact  that  the  drier  Western 
states  have  been  settled. 

We  proceed  in  a  similar  way  when  computing  the  mor- 
tality index  which  is  found  for  a  certain  population  on 
the  basis  of  the  age  and  sex  classification  of  a  standard 
population.  A  weighted  arithmetic  mean  is  computed  from 
the  special  death  rates  for  the  different  classes  of  age  or 

*^  "  Some  Statistics  Relating  to  Working  Class  Progress  since 
1860,"  Journ.  of  the  Roy.  Stat.  Soe.,  Vol.  LXII  (1899),  especially  p. 
655  fiF. 


160  THE  VARIOUS  KINDS  OF  AVERAGES 

sex  of  the  population  by  combining  these  special  death 
rates  with  weights  which  belong  to  them  according  to  the 
age  and  sex  classification  of  the  standard  population. 
The  purpose  of  the  computation  of  mortality  indices  for 
different  countries  on  the  basis  of  the  same  standard  popu- 
lation is,  of  course,  to  find  indices  which  are  comparable 
for  all  countries  independent  of  the  different  age  or  sex 
constitution  of  the  respective  populations.^^ 

A  counterpart  to  the  method  of  the  standard  population 
is  the  method  of  the  standard  mortality .^^  According  to 
Professor  von  Bortkiewicz  it  is  "  the  comparison  between 
the  number  of  deaths  actually  occurring  and  the  number 
of  deaths  expected  to  occur  according  to  a  standard  mor- 
tality." To  the  general  death  rate  computed  in  the  normal 
way  which  a  certain  population  shows  on  the  basis  of  its 
age  constitution  and  the  mortality  conditions  in  the  various 
age  classes,  is  contrasted  the  mortality  rate  which  the  same 
population  would  show  if  in  its  various  age  classes  those 
mortality  conditions  were  prevalent  that  are  found  in  the 
population  chosen  as  a  standard.  This  standard  mortality 
rate  with  which  the  actual  death  rate  of  a  country  is  to 
be  compared,  is  found  by  computing  a  weighted  arithmetic 
mean  from  the  special  death  rates  for  the  single  age  classes 
of  the  standard  population.  In  this  operation  every  one 
of  these  death  rates  is  given  that  weight  which  belongs  to 
corresponding  age  class  according  to  the  age  constitution 
of  the  concrete  population  in  question.  This  method  plays 
an  important  role  in  the  practice  of  life  insurance  com- 

"^  Compare  especially  "  Mortalitats-KoefRzient  und  MortalitSts- 
Index"  in  the  Bull,  de  I'Inst.  intern,  de  Stat.,  Vol.  VI,  No.  2,  and 
"  tJber  die  Berechnung  eines  internationalen  Sterbliehkeitsmassea 
(Mortality-Index)  "  in  Conrad's  Jalirbiieher,  3rd  series,  Vol.  VI 
(1893),  by  Josef  Korosi,  as  well  as  the  works  of  Ogle,  Rubin,  Sund- 
baerg,  and  v.  Bortkiewicz. 

"Compare  Uber  die  Methode  der  "Standard  Population"  by  Dr. 
L.  V.  Bortkiewicz,  Berlin,  1903  (Reports  of  the  9th  Session  of  the 
International  Statistical  Institute). 


THE  ARITHMETIC  MEAN  161 

panies,  but  so  far  has  found  little  use  in  general  population 
statistics.^* 

Certain  statisticians  have  noted  that,  in  certain  cases, 
the  use  of  weights  has  only  very  insignificant  influence  upon 
the  numerical  value  of  the  arithmetic  mean,  so  that,  with 
or  without  the  use  of  weights,  and  also  with  the  use  of 
different  weights,  we  often  obtain  almost  the  same  mean. 
This  observation  was  made  especially  in  the  computation 
of  weighted  arithmetic  means  from  series  of  price  indices 
and  it  has  been  discussed  by  Giffen,  Sauerbeck,  Taussig, 
and  others.  The  use  of  weights  in  the  computation  of  mean 
index  numbers,  which  is  a  laborious  process  and  continually 
leads  to  controversies,  appeared  to  be  superfluous,  and  the 
computation  of  simple  arithmetic  means  from  indices  of 
prices  seemed  to  be  justified.^^ 

However,  experience  gathered  from  isolated  cases  and 
under  certain  conditions  must  not  be  immediately  general- 
ized. If  only  a  few  items  of  very  different  weights  are 
given,  then,  of  course,  we  obtain  decidedly  different  values 
for  simple  and  weighted  arithmetic  means.  In  such  a  case 
we  certainly  cannot  do  without  the  use  of  weights.  But 
if  a  large  number  of  items  is  given,  then  it  is  quite  possible 

**  The  method  of  the  expected  deaths — as  the  most  important  appli- 
cation of  the  more  general  "  Jlethod  of  expected  events  " — was  advo- 
cated especially  by  Westergaard  (cf.  his  "  Alte  und  neue  Messungs- 
vorschliige  in  der  Statistik"),  Conrad's  Jahrbiicher,  3rd  series,  Vol. 
VI  (1893),  p.  330  flf.  Bleicher  has  also  concerned  himself  with  this 
method   (cf.  v.  Mayr,  Bevolkerungsstatistik,  p.  220). 

*°  In  his  work  in  the  field  of  historical  wage  statistics  Bowley  has 
found  that  weights  eventually  have  only  small  influence  on  the  size 
of  the  arithmetic  mean  (cf.  Economic  Journal,  Vol.  V,  p.  373,  and 
Joum.  of  the  Roy.  Stat.  Soc,  Vol.  LXII  (1899),  p.  712,  and  Vol. 
LXIX  (1906),  p.  164  flF.).  Wood  obtained  a  similar  result  in  his 
investigations  on  the  development  of  English  consumption,  where  he 
computed  and  compared  simple  arithmetic  means  and  arithmetic 
means  weighted  according  to  5  difTerent  systems  (cf.  "Some  Sta- 
tistics Relating  to  Working  Class  Progress  since  1860,"  Journ.  of  the 
Roy.  Stat.  Soc,  Vol.  LXII   (1899),  particularly  p.  655  ff.). 


162  THE  VARIOUS  KINDS  OF  AVERAGES 

that  the  weights  of  the  measurements  above  the  average 
approximately  balance  the  ones  below  the  average. ^^*  The 
use  of  weights  in  such  a  case  is  without  effect,  since  the 
weights  neutralize  each  other  in  the  computation  of  the 
mean.  If  there  is  no  relationship  between  the  numerical 
size  of  the  items  and  the  weights  which  belong  to  them, 
then  it  is  probable  that  the  weights  of  items  above  and  below 
the  average  are  approximately  equal  and  thus  have  no 
noticeable  effect.  In  this  way  the  fact  that  weights  fre- 
quently have  hardly  any  influence  upon  the  arithmetic 
mean  may  be  explained. ^^ 

However,  the  weights  of  the  items  must  not  be  neglected 
if  there  exists  a  close  connection  between  the  numerical 
value  and  the  weight  of  the  items.  Bowley  gives  a  striking 
example  of  this.  Suppose  we  desire  to  compute  the  average 
wage  for  a  whole  country  from  the  rates  paid  to  workmen 
of  a  certain  occupation  in  the  different  cities  of  the  country. 
We  may  take  the  rates  found  in  the  different  cities  to  be  of 
equal  weight  and  compute  a  simple  arithmetic  mean.  But 
we  may  also  take  into  account  that  the  number  of  work- 
men belonging  to  the  occupation  is  different  in  the  different 
cities,  and  use  the  corresponding  numbers  as  weights  in  the 
computation  of  the  mean.  The  latter  method  of  computa- 
tion will  result  in  a  considerably  higher  average  number, 

"a  Thus,  the  United  States  Labor  Bureau  used  the  simple  average 
in  computing  the  general  index  number  of  wholesale  prices  of  some 
250  commodities  and  the  Canadian  Department  of  Labor  likewise 
used  the  simple  average  of  the  indices  of  wholesale  prices  of  230 
articles.  In  computing  an  index  of  retail  prices  the  United  States 
Bureau  of  Labor  uses  both  the  simple  average  of  thirty  articles  of 
food  and  the  weighed  average  in  which  the  weights  are  chosen 
according  to  the  average  family  consumption  as  shown  in  2,567 
budgets.  The  average  absolute  difference  between  the  simple  and 
weighted  averages  for  the  years  1890-1906  is  0.33;  the  difference 
exceeds  0.6  in  but  one  year,  1900,  when  it  is  1.4.  (See  Bulletin 
of  the  Bureau  of  Labor  No.  77  for  retail  prices,  1890  to  1907,  and 
Bulletin  No.  87  for  wholesale  prices,  1890-1910.) — Translator. 

'"Bowley,  Elements  of  Statistics,  2nd  ed.,  p.  117. 


THE  ARITHMETIC  MEAN  163 

as  there  is  a  close  connection  between  the  amount  of  wages 
and  the  number  of  workmen.  In  the  larger  cities,  with 
a  greater  number  of  workmen  belonging  to  an  occupation, 
the  wages  are  generally  higher  and  hence  the  weighted  and 
simple  arithmetic  means  must  differ  considerably. 

The  question,  whether  weights  are  to  be  used  or  not, 
cannot,  therefore,  be  decided  in  general.  The  answer  de- 
pends on  the  question  of  the  relationship  between  the  items 
and  their  weights.  This  latter  question,  however,  cannot 
always  be  decided  a  priori.  Therefore  it  is  sometimes  nec- 
essary to  experiment  with  weights.  If  by  using  them  we 
find  a  value  which  does  not  differ  essentially  from  the 
simple  arithmetic  mean,  then  the  use  of  weights  evidently 
has  no  practical  importance.  But  if  the  weighted  arith- 
metic mean  differs  considerably  from  the  simple  arithmetic 
mean,  then  as  a  rule  the  consideration  of  the  weight  of  the 
items  cannot  be  neglected. ^''^ 


C.    THE    ARITHMETIC   MEAN   AND    MATHEMATICAL 
STATISTICS 

The  arithmetic  mean  is  widely  used  in  the  theory  of 
error  and  the  theory  of  probability,  the  principles  of  which 

*°aA  committee  of  the  British  Association  was  appointed  to  in- 
vestigate the  question  of  price  index  numbers.  Sir  Robert  Giffen 
states  the  conclusion  of  the  committee  as  follows  (Report  of  the 
Brit.  Assoc,  1888,  p.  184;  also  quoted  in  article  on  "Index  Num- 
bers" in  Palgrave's  Diet.  Pol.  Econ.)  :  "The  articles  as  to  which 
records  of  prices  are  obtainable  being  themselves  only  a  portion  of 
the  whole,  nearly  as  good  a  final  result  may  apparently  be  arrived 
at  by  a  selection  without  bias,  according  to  no  better  principle  than 
accessibility  of  record,  as  by  a  careful  attention  to  weighting.  .  .  . 
Practically  the  committee  would  recommend  the  use  of  a  weighted 
index  number  of  some  kind,  as,  on  the  whole,  commanding  more 
confidence.  ...  A  weighted  index  number,  in  one  respect,  is  almost 
an  unnecessary  precaution  to  secure  accuracy,  though,  on  the  whole, 
the  Committee  recommended  it." — Teanslator, 


164  THE  VARIOUS  KINDS  OF  AVERAGES 

have  been  applied  to  statistical  series  and  means.^'^  The 
most  important  of  these  principles  will  be  given  as  briefly 
as  possible,  with  reference  to  the  literature,  and  the  conse- 
quences arising  from  the  application  of  these  principles 
to  statistical  data  will  be  pointed  out.  "While  doing  this 
we  must  distinguish  between  series  of  quantitative  indi- 
vidual observations  and  series  of  statistical  items,  each  of 
which  expresses  some  nmnerieal  probability. 


1.  THE  ARITHMETIC  MEAN  OP  SERIES  OF  QUANTITATIVE  INDI- 
VIDUAL OBSERVATIONS  AND  THE  THEORY  OF  ERRORS  OF 
OBSERVATION 

In  order  to  judge  the  reliability  of  the  arithmetic  mean 
of  a  series  of  quantitative  individual  observations  from 
the  standpoint  of  mathematical  statistics,  we  must  go  back 
to  the  theory  of  errors  of  observation.  It  is  a  fact,  based 
upon  experience,  that  repeated  measurements  of  an  object, 
the  size  of  which  is  to  be  found,  do  not  completely  coincide. 
The  reason  for  this  lies  in  the  fact  that  neither  the  human 
senses  nor  the  measuring  instruments  are  absolutely  accu- 
rate. Therefore,  the  individual  measurements  are  affected 
with  accidental  errors   of  observation.     These  accidental 

"  As  the  principal  representatives  of  mathematical  statistics  may 
be  mentioned:  Lexis,  Edgeworth,  v.  Bortkiewicz,  West^rgaard,  Gal- 
ton,  Pearson,  Yule,  Bowley,  Fechner,  Czuber,  and  Blaschke.  J.  v. 
Kries  has  extended  the  application  of  the  calculus  of  probability  to 
statistics,  especially  on  the  logical  and  the  perceptive  theoretical 
sides.  G.  F.  Knapp  is  the  most  prominent  opponent  of  the  applica- 
tion of  the  calculus  of  probability  to  statistical  investigation  (ef. 
his  articles,  "Die  neueren  Ansichten  iiber  Moral statistik"  and 
"Quetelet  als  Theoretiker,"  in  Conrad's  Jahrbiicher,  Vols.  XVI- 
XVIII,  1871-1872).  A.  M.  Guerry  also  expressed  himself  against  the 
use  of  the  calculus  of  probability  in  statistics  (Statistique  morale  de 
I'Angleterre  comparee  avec  la  statistique  morale  de  la  France,  Paris, 
1864,  p.  xxxiii  ff.). 


THE  ARITHMETIC  MEAN  165 

errors  are  partly  positive,  partly  negative,  and  we  know 
from  experience  that  smaller  accidental  errors  occur  more 
frequently  than  larger  ones.  Moreover,  according  to  the 
theory  of  errors  of  observation,  numerous  measurements  of 
the  same  object  group  themselves  around  their  arithmetic 
mean  according  to  the  Gaussian  law  of  accidental  errors, 
and  this  mean  may  be  considered  to  be  the  most  probable 
value  of  the  quantity  measured.  The  true  value  of  the 
quantity  under  observation  cannot  be  determined.  But 
superior  and  inferior  limits  can  be  found,  between  which 
the  true  value  lies  with  a  given  numerical  probability. 
These  limits  may  be  made  to  approach  each  other  in  two 
ways,  first  by  procuring  more  accurate  instruments,  since 
the  accuracy  of  the  arithmetic  mean  of  a  number  of  ob- 
servations depends  upon  the  accuracy  of  the  single  observa- 
tions, second  by  increasing  the  number  of  observations, 
for  the  precision  of  the  arithmetic  mean  of  a  number  of 
measurements  varies  directly  with  the  square  root  of  the 
number  of  measurements.  Therefore,  in  order  to  obtain  a 
result  twice  or  three  times  as  precise,  we  have  to  make 
four  or  nine  times  as  many  observations.  The  greater  the 
number  of  observations,  the  greater  will  be  the  accuracy 
of  their  arithmetic  mean  and,  therefore,  the  greater  the 
probability  that  the  true  value  is  between  given  limits 
above  and  below  this  mean.  Or,  stating  the  same  idea 
differently,  the  greater  the  number  of  observations  the 
nearer  together  are  the  limits  between  which  the  true  value 
is  contained  with  given  probability.  With  an  infinite  num- 
ber of  observations  their  arithmetic  mean  ought  to  repre- 
sent the  true  value  of  the  measured  quantity.  The  accuracy 
of  the  determination  of  the  arithmetic  mean,  called  its 
'  *  precision, ' '  may  be  expressed  numerically  according  to  a 
certain  formula.  The  reciprocal  of  the  precision,  the 
"  modulus  "  (the  square  of  which  is  called  the  "fluctua- 
tion "),  the  mean  error,  and  the  probable  error  of  the  mean 
may  all  be  used  as  measures  of  accuracy.     They  are  con- 


166  THE  VARIOUS  KINDS  OF  AVERAGES 

nected  by  mathematical  formula  so  that  one  may  be  com- 
puted from  the  other.^^* 

"a  Suppose  that  a  very  large  number  of  measurements  of  a  single 
physical  quality  are  taken.  Suppose  further  that  our  measuring 
instrument  is  so  adjusted  that  there  is  no  uniform  tendency  to  give 
too  large  or  too  small  values;  in  other  words,  to  give  systematic  errors. 
However,  there  will  be  variations  in  the  resulting  measurements; 
the  measurements  will  be  affected  by  accidental  errors.  Suppose 
that  we  make  the  following  assumptions: 

(i)  That  the  arithmetic  mean  of  the  observations  is  the  most 
probable  value  of  the  quality  that  is  being  measured; 

(ii)   That  positive  and  negative  errors  are  equally  probable; 

(iii)  That  small  errors  are  relatively  more  probable  than  large 
ones; 

(iv)   And  that  the  contributory  causes  of  error  are  independent. 

If  we  let  X  stand  for  the  accidental  error  (the  difference  between 
any  observation  and  the  arithmetic  mean  of  all  observations)  then 
the  law  of  distribution  of  errors  or,  in  other  words,  the  frequency- 
curve  of  error  is: 

f(x)=_^.  e-i>'x» 

where 

e=z  2.71828   .    .    . 
,r  =  3.14159    .    .    . 

h=4/-^?^ — z=.th.e  precision. 
r    22x2 

n  =z  number  of  observations. 
S  z=:  "  sum  of  such  terms  as  " 

—  :=  modulus. 
h 

0.4769363^  p^^^^^;^  ^^^^^^ 
SIxl        1 


mean  error. 


(|x|    indicates  that  all  errors  are  considered  positive.) 


— z=  fluctuation. 


— Tbansiatob, 


THE  ARITHMETIC  MEAN  167 

Therefore,  the  arithmetic  mean  of  a  series  of  repeated 
measurements  of  the  same  object  has  a  special  scientific 
importance  in  the  theory  of  errors.  It  is  only  by  com- 
puting the  arithmetic  mean  from  the  measurements  that 
the  true  value  of  the  measured  object  which  is  of  primary 
importance,  can  be  closely  approximated.  Every  single 
measurement  may  show  an  error  of  unknowTi  quantity.  The 
theory  of  errors  is  of  especial  value  in  astronomy  and 
geodetic  survey.  In  order  to  find  the  true  size  of  an 
object,  arithmetic  means  are  computed  from  numerous 
measurements  of  it. 

In  statistics  it  is  not  a  question  of  repeated  measurements 
of  the  same  object,  but  of  measurements  of  different  similar 
objects,  each  being  measured  but  once.  Now  it  has  been 
noticed  that  statistical  series  sometimes  show  the  same  dis- 
tribution around  their  arithmetic  mean  as  repeated  obser- 
vations of  the  same  object.  This  distribution,  which  cor- 
responds to  the  Gaussian  law,  is  especially  characterized 
by  the  facts  that  the  single  observations  are  grouped  sym- 
metrically around  their  arithmetic  mean,  and  that  the  items 
are  densest  around  the  arithmetic  mean  and  become  rarer 
the  farther  they  deviate  from  it.  On  account  of  the  con- 
centration and  symmetrical  distribution  of  the  items  around 
their  arithmetic  mean,  the  mode  and  the  median  of  the 
series  theoretically  coincide.  Practically,  however,  they 
usually  differ  somewhat  because  of  the  comparatively  small 
number  of  observations  at  hand. 

A  statistical  series,  the  items  of  which  are  distributed 
around  the  arithmetic  mean  according  to  the  Gaussian 
law,  has  the  appearance  of  a  series  of  repeated  measure- 
ments of  the  same  object.  It  is  an  obvious  step,  therefore, 
to  apply  the  theorems  resulting  from  the  theory  of  errors 
of  observation  to  such  statistical  series  and  their  means. 
Mathematical  statistics  first  investigates  the  distribution  of 
the  items  of  statistical  series  around  the  arithmetic  mean. 
In   case  series  follow  the   Gaussian  law  they   are   called 


168  THE  VARIOUS  KINDS  OF  AVERAGES 

"  typical  "  series,  and  the  means  from  such  series  "  typ- 
ical "  means.  If  a  "  typical  "  arithmetic  mean  is  given, 
its  accuracy  is  ascertained  and  the  dispersion  of  the  series 
around  the  mean  is  measured  in  the  same  way  as  though 
it  were  a  question  of  mere  errors  of  observation  with  re- 
peated measurements  of  one  and  the  same  object. 

Obviously,  the  ideas  of  the  theory  of  errors  cannot  be 
applied  to  typical  statistical  means  without  certain  changes. 
The  arithmetic  mean  of  a  series  of  measurements  of  differ- 
ent objects  (measurements  of  human  height,  length  of  life, 
wages)  cannot  be  considered  to  be  the  most  probable  value 
of  the  "  true  "  size  of  a  certain  object,  since  all  the  single 
units  of  observation  are  independent  real  phenomena  and 
equally  "  true."  However,  we  may  take  the  arithmetic 
mean  of  the  measurements  to  be  a  normal  value  (or  the 
most  probable  empirical  determination  of  a  theoretical 
normal  value)  the  size  of  which  is  determined  by  the 
general  common  causes  influencing  all  the  units  of  observa- 
tion and  from  which  individual  cases  differ  merely  on 
account  of  the  disturbance  due  to  individual  accidental 
causes.  Accordingly  the  arithmetic  mean  represents  the 
"  type  "  of  the  observed  phenomenon  which  is  expressed 
with  merely  accidental  variations  in  the  individual  cases. 
From  this  argument  the  great  scientific  importance  which 
mathematical  statisticians  attribute  to  "  typical  "  averages 
can  be  recognized.  The  typical  means  are  independent 
scientific  perceptions.  The  series  of  items  compared  with 
the  typical  mean  thus  loses  the  greatest  part  of  its  im- 
portance. It  is  worthy  of  consideration  only  as  a  measure- 
ment of  the  variability  of  the  type. 

The  application  of  the  theory  of  errors  of  observation 
to  statistical  series  of  measurements  undoubtedly  has  great 
theoretical  value.  Its  practical  importance,  however,  is 
small,  since  statistical  series  of  measurements  which  con- 
form to  the  Gaussian  curve  occur  only  very  rarely.  A 
number  of  such  series  were  found  in  anthropometry,  and 


THE  ARITHMETIC  MEAN  169 

Quetelet  thought  that  anthropometric  measurements,  espe- 
cially height,  were  generally  distributed  symmetrically 
around  their  arithmetic  mean  according  to  the  law  of 
errors.  More  recent  investigations,  however,  have  estab- 
lished other  forms  of  distribution  in  most  cases.  If  statis- 
tical series  show  any  regularity  at  all,  it  agrees  only  in 
very  rare  exceptions  with  the  normal  Gaussian  curve.  As 
a  rule  regular  conformations  of  a  different  kind  occur, 
such  as  the  asymmetrical  Gaussian  curve  emphasized  by 
Fechner,  or  the  skew  curve  of  error.  But  in  such  cases 
the  application  of  the  principles  of  the  theory  of  errors  of 
observation  to  statistics  loses  a  great  deal  of  its  importance. 
This  theory  cannot  be  applied  to  means  of  series  that  do 
not  follow  the  Gaussian  law.  In  such  series  arithmetic 
mean,  mode,  and  median  are  separate  values  and  often 
differ  considerably,  and  there  is  no  "  normal  value,"  in 
the  strictly  mathematical  sense,  from  which  the  single  items 
differ  only  by  accidental  deviations.  There  is  no  "  typ- 
ical "  mean  which  could  be  used  to  stand  for  the  series  as 
a  whole.  But  still  the  arithmetic  means  of  series  which  do 
not  coincide  with  the  normal  Gaussian  curve  are  by  no 
means  inadmissible  or  unimportant.  On  the  contrary  the 
arithmetic  mean  offers  valuable  information  about  the  series 
in  question,  even  if  it  lacks  the  special  justification  found 
in  the  theory  of  errors.  It  characterizes  the  series,  as  has 
been  shown  above,^®  in  many  important  points. 

The  mathematical-statistical  argument  is  based  on  an 
analogy  between  statistical  series  of  once-occurring  meas- 
urements of  separate  but  similar  objects  and  repeated 
measurements  of  the  same  object,  customary  in  geodetics 
and  astronomy.  In  this  the  deviations  from  the  average 
which  the  items  of  a  statistical  series  (human  heights  or 
lengths  of  life)  show,  are  placed  on  a  par  with  "  acciden- 
tal "  errors  of  observation  and  are  treated  according  to 

"  See  the  section  "  Concept  and  Qualities  of  the  Arithmetic  Mean," 
pp.  138  ff. 


170  THE  VARIOUS  KINDS  OF  AVERAGES 

the  mathematical  theorems  developed  for  their  treatment. 

The  principles  of  the  theory  of  errors,  however,  may  be 
applied  to  series  of  statistical  measurements  in  still  another 
way.  In  this  application  we  do  not  place  the  deviations 
from  the  average  on  a  par  with  errors  of  observation. 
On  the  contrary,  we  examine  the  actual  errors  of  observa- 
tion of  each  of  the  items  and  investigate  the  connection 
existing  between  these  errors  and  the  error  of  the  average. 
For  errors  of  observation  are  connected  not  only  with 
astronomical  measurements  but,  in  most  cases,  also  with 
statistical  measurements. 

The  errors  which  occur  in  statistical  measurements  may 
be  like  the  errors  with  repeated  measurements  of  the  same 
object,  either  systematic  errors  (biassed  errors)  or  acciden- 
tal errors  (unbiassed  errors).  Systematic  errors  are  all 
made  in  the  same  direction.  If  the  construction  of  a  physi- 
cal instrument  is  incorrect,  then  all  the  single  measure- 
ments show  the  same  systematic  error,  and  the  error  ap- 
pears in  the  average.  Systematic  errors  frequently  occur 
in  statistics.  If  many  women,  out  of  vanity,  have  stated 
their  ages  too  low  in  the  census,  then  this  is  a  systematic 
error  adhering  to  the  age  data  which  must  be  reflected  in 
the  average  age  of  women.  The  error  of  the  average  age 
is  equal  to  the  average  error  of  the  items.  A  systematic 
error  will  also  occur  in  an  investigation  of  supposedly  repre- 
sentative cases  (not  a  complete  census)  if  the  choice  of 
cases  is  ruled  by  criteria  which  tend  to  make  them  deviate 
in  the  same  direction.  For  instance,  wage  data  ascer- 
tained in  a  representative  investigation  often  come  mainly 
from  the  larger  and  more  prominent  factories  where  higher 
wages  may  be  paid  than  in  smaller  factories.  The  average 
wage  evidently  must  be  higher  than  if  all  the  factories 
had  been  taken  into  account  uniformly. 

Accidental  errors,  as  distinguished  from  systematic,  are, 
as  a  rule,  partly  positive,  partly  negative.  They  form  the 
subject-matter  of  the  theory  of  errors.     The   arithmetic 


THE  ARITHMETIC  MEAN  171 

mean  possesses  the  property  that  in  its  computation  the 
accidental  errors  of  observation  adhering  to  the  single 
observations  neutralize  each  other,  the  more  completely  the 
greater  the  number  of  observations.  This  property  origi- 
nally proven  for  repeated  observations  of  the  same  object 
also  holds  for  errors  connected  with  single  observations  of 
different  but  similar  units.  Also  in  this  case  the  error  of 
the  average  is  considerably  smaller  than  the  probable  error 
of  a  single  observation,  and  the  accuracy  of  the  average 
varies  directly  with  the  square  root  of  the  number  of  ob- 
servations.^^ The  median  and  mode  have  no  similar  prop- 
erty. These  means  are  obtained  by  selecting  certain  items 
as  being  characteristic  of  the  whole  series  and,  therefore, 
they  share  the  errors  of  observation  of  concrete  items. 

2.    THE    ARITHMETIC    MEAN    OP    SERIES    OF    STATISTICAL 
PROBABILITIES  AND  THE  THEORY  OF  PROBABILITY 

Just  as  the  theory  of  errors  of  observation,  under  definite 
conditions,  gives  a  reason  for  using  the  arithmetic  mean 
of  statistical  measurements,  so  the  theory  of  probability, 
likewise  under  definite  conditions,  provides  a  reason  for 
adopting  the  arithmetic  mean  of  series  of  statistical  proba- 
bilities. The  reasoning  of  mathematical  statisticians,  which 
is  shortly  explained  in  the  following,  is  based  upon  the 
law  of  great  numbers  formulated  by  Bernoulli  and  Poisson, 
mainly  with  regard  to  experiences  in  games  of  chance,  or 
rather  upon  the  inversion  of  this  law  (theorem  of  Bayes) . 

If  a  certain  theoretical  probability  exists  that  an  event 
is  going  to  happen,  then  an  approximation  to  this  probabil- 

••  Bowley,  especially,  has  examined  carefully  the  connection  be- 
tween the  accuracy  of  a  statistical  average  and  the  accuracy  of  the 
items  on  which  the  former  is  based.  (Cf.  "Relations  between  the 
Accuracy  of  an  Average  and  That  of  Its  Constituent  Parts  "  in  Journ. 
of  the  Roy.  Stat.  Soc,  Vol.  LX  (1S97),  p.  855  ff.,  and  Elements  of 
Statistics,  Chap.  VIII,  "Accuracy.") 


172  THE  VARIOUS  KINDS  OF  AVERAGES 

ity  may  be  found  by  experiment.  That  is,  the  empirical 
probability  originating  from  experiment  usually  agrees 
approximately,  but  as  a  rule  not  completely,  with  its  theo- 
retical probability.  If  by  continued  experiment  several 
empirical  probabilities  (i.  e.,  empirical  values  affected  with 
merely  accidental  errors)  are  obtained  for  the  same  theo- 
retical probability,  then  these  fluctuate  symmetrically  within 
certain  limits  around  such  theoretical  probability.  The 
arithmetic  mean  of  the  empirical  probabilities  is  the  nearest 
approximation  to  the  theoretical  probability  and  may  be 
considered  to  be  its  most  probable  value. 

The  relation  existing  between  the  empirical  frequency 
of  an  event  and  its  theoretical  probability  obeys  the  law 
of  great  numbers.  The  greater  the  number  of  observa- 
tions upon  which  the  empirical  frequency  is  based  the 
more  closely  does  this  frequency  follow  the  theoretical 
probability.  The  greater  the  number  of  experiments,  the 
greater  is  the  probability  that  the  difference  between  the 
empirical  frequency  of  the  event  in  question  and  its 
theoretical  probability  is  within  assigned  limits,  or,  stated 
in  another  way,  the  narrower  are  the  limits  within  which 
the  difference  mentioned  lies  with  assigned  probability. 
The  degree  of  accuracy  with  which  an  empirical  value 
corresponds  to  the  theoretical  probability  may,  therefore, 
be  expressed  numerically  in  the  individual  case  by  the 
"  precision  "  of  the  empirical  value,  or  by  the  reciprocal 
of  the  precision,  the  "  modulus,"  or  by  the  mean,  the 
average,  or  the  probable  error  of  the  empirical  value.  If 
several  values  of  an  empirical  probability  are  given  for 
the  same  theoretical  probability,  then,  according  to  the 
law  of  great  numbers,  their  deviations  from  the  theoretical 
probability  vary  inversely  with  the  number  of  experiments 
on  which  they  are  based.  In  a  given  case,  the  probability 
of  a  given  deviation  from  the  theoretical  probability  can 
be  computed. 

Let  us  illustrate  this  by  an  example.     If  a  drawing 


THE  ARITHMETIC  MEAN  173 

is  made  from  an  urn  containing  red  and  white  balls  in 
the  proportion  6 : 4,  then  there  is  a  theoretical  probability 
of  0.6  that  a  red  ball  will  be  drawn,  a  theoretical  probability 
of  0.4  that  a  white  ball  will  appear.  Now  let  us  make 
experiments  by  drawing  a  ball  from  the  urn  and  putting 
it  back  1,000  times  in  succession.  It  is  very  probable 
that  the  ball  will  not  be  drawn  in  the  ratio  of  600  to  400, 
but  empirical  probabilities  for  the  red  or  white  balls  will 
be  found  which  deviate  slightly  from  the  theoretical  proba- 
bilities of  both  colors.  If  we  repeat  this  process  and  ascer- 
tain the  percentages  of  the  red  and  the  white  balls  for 
several  successive  drawings  of  1,000  balls  each,  then  we  find 
for  each  color  a  series  of  empirical  probabilities  (i.  e.,  em- 
pirical values,  affected  with  merely  accidental  errors)  which 
fluctuate  symmetrically  within  certain  limits  around  the 
theoretical  probability.  The  theory  of  probability  enables 
us  to  express  numerically  the  degree  of  accuracy  of  the 
empirical  values  and  to  compute  a  priori  the  probability 
with  which  different  deviations  from  the  theoretical  prob- 
ability are  to  be  expected.  In  two  out  of  three  drawings 
of  1,000  balls  the  number  of  the  red  would  deviate  at  most 
2.6^  from  the  true  number,  i.  e.,  the  empirical  value  would 
be  between  0.616  and  0.584,  only  very  rarely  there  would 
occur  a  deviation  in  which  the  number  of  the  red  balls 
would  be  greater  than  650  or  smaller  than  550.^° 

If  we  group,  not  1,000,  but  100,000  successive  drawings 
from  the  urn,  and  if  we  find  the  percentage  of  the  red  and 
white  balls  drawn,  then  we  obtain  for  each  color  an  em- 
pirical probability  which,  according  to  the  law  of  great 
numbers,  probably  more  closely  approaches  the  theoretical 
probability  than  an  empirical  probability  based  on  only 
1,000  drawings.  Therefore,  a  series  of  values  of  empirical 
probability,  each  based  on  100,000  drawings,  will  fluctuate 
around  the  theoretical  probability  within  relatively  nar- 

*"  Cf.  Harold  Westergaard,  Die  Grundziige  der  Theorie  dcr  Sta- 
tistik,  p.  57. 


174  THE  VARIOUS  KINDS  OF  AVERAGES 

rower  limits  than  a  series  of  analogous  values  each  based 
on  only  1,000  drawings. 

Thus,  the  law  of  great  numbers  explains  what  empirical 
values  may  be  expected  with  assigned  probabilities,  if  ex- 
periments are  made  which  we  base  on  a  known  theoretical 
probability.  On  the  other  hand,  the  inverse  of  this  theorem 
permits  us,  under  certain  conditions,  to  draw  conclusions 
from  given  observations  to  the  unknown  theoretical  proba- 
bility on  which  these  observations  are  based.  It  is  this 
inversion  of  Bernoulli's  theorem  which  can  be  used  in 
statistics.  Statistical  events  show  certain  analogies  to  acci- 
dental events.  The  various  statistical  events  (births, 
deaths,  etc.)  apparently  occur  with  the  same  irregularity 
as  the  drawings  of  red  and  white  balls  from  an  urn.  How- 
ever, great  regularity  is  found  when  many  individual  events 
are  combined.  Mathematical  statisticians,  therefore,  fre- 
quently conceive  relative  numbers  fulfilling  certain  con- 
ditions (statistical  probabilities)  to  be  empirical  determina- 
tions of  (unknown)  theoretical  probabilities  or  of  functions 
of  such.*^  From  the  empirical  values  conclusions  are  made 
as  to  the  more  important  theoretical  probability.  It  is 
possible  to  determine  between  what  limits  above  and  below 
the  empirical  value  the  theoretical  probability  is  situated 
with  any  assigned  probability.*^ 

"  Corresponding  to  the  average  character  of  most  of  the  relative 
numbers  ( cf .  above,  p.  38  ff . )  it  can,  as  a  rule,  be  assumed  that  they 
do  not  express  a  single  uniform  probability,  but  an  "  average  prob- 
ability," so  that  particular  "  special  probabilities  "  exist  for  certain 
constituents.  The  consequences  which  result  from  the  application  of 
the  theory  of  probability  to  statistical  probabilities,  have  been  treated 
thoroughly  by  L.  v.  Bortkiewicz  in  his  paper,  "  Kritische  Betrach- 
tungen  zur  theoretischen  Statistik,  I.  Artikel "  ( Conrad's  Jahr- 
biicher,  3rd  series,  Vol.  VIII  (1894),  p.  641  f.).  These  consequences 
do  not  bear  upon  what  is  said  in  the  following  text. 

*' For  this  Czuber  gives  the  following  example  (Wahrscheinlich- 
keitsrechnung,  p.  304)  :  of  54,391  males  who  completed  the  age  of 
50  years,  according  to  the  German  mortality  tables  of   1883,   1,049 


THE  ARITHMETIC  MEAN  175 

According  to  Bernoulli's  theorem  these  limits  are  closer 
together  the  greater  the  number  of  observations.  If  we 
find,  for  instance,  2,600  male  births  in  a  total  of  5,000,  then 
the  theoretical  probability  of  the  birth  of  a  boy  is  deter- 
mined only  within  rather  widely  separated  limits.  This 
relative  number  may  indicate  that  the  theoretical  proba- 
bility is  located  between  |^-^^  and  |H§.  But  if  we  have 
observed  500,000  births  in  which  there  are  260,000  males, 
then  we  may  expect  that  the  observed  number  differs  only 
1,000  at  most  from  the  theoretically  most  probable  number ; 
therefore  this  would  be  located  between  259,000  and  261,- 
000.*3 

The  mathematical  theorems  mentioned  evidently  empha- 
size the  importance  of  the  arithmetic  mean.  If  a  series 
of  relative  numbers  be  given  which  can  be  considered  to 
be  empirical  probabilities  (for  instance,  a  series  of  relative 
numbers  which  indicate  the  number  of  male  births  in  pro- 
portion to  the  total  number  of  births  by  years  or  dis- 
tricts), then  the  arithmetic  mean  of  these  numbers  is  based 
on  a  much  larger  number  of  observations  than  any  in- 
dividual relative  number.  Therefore,  according  to  the 
theory  of  probability  it  approaches  the  true  theoretical 
probability  with  which  the  mathematical  statisticians  are 
concerned,  considerably  nearer  than  do  the  individual  rela- 
tive members.  It  may  be  considered  to  be  the  most  proba- 
ble value  of  the  theoretical  probability.  Thus  the  arith- 
metic mean  possesses  considerably  greater  scientific  value 
than  the  items. 

However,  the  application  of  the  law  of  great  numbers, 

died  before   they   reached   the   age   of  51.     Tlie  empirical  value   of 

543^1  ~  ^•^^^^^'  '^^^^  ^^^  probable  error  of  0.000397,  is  obtained  from 

these  data  for  the  probability  of  der.th  of  the  50-year-old  males, 
so  that  an  even  bet  could  be  made  that  the  probability  mentioned 
falls  between  the  limits  0.01 889  and  0.01969. 

*•  Westergaard,  Die  Grundziige  der  Theorie  der  Statistik,  p.  57  f. 


176  THE  VARIOUS  KINDS  OF  AVERAGES 

or  its  inverse,  to  statistical  relative  numbers  is  feasible 
only  under  certain  assumptions.  The  theorems  of  the 
theory  of  probability  can  only  be  applied  to  relative  num- 
bers which,  in  a  formal  way,  can  be  considered  to  be 
probabilities  or  functions  of  such,**  However,  as  is  known, 
practical  statistics  has  frequently  to  do  with  relative  num- 
bers that  are  not  of  such  form.  All  the  frequency  numbers 
and  coefficients  which  originate  by  correlating  certain  events 
(births,  deaths,  marriages,  crimes,  etc.)  with  the  average 
population  are  neither  probabilities  nor  functions  of  such. 
And  yet  these  values  (birth  rates,  death  rates,  marriage 
rates)  form  the  chief  material  of  practical  statistics,  while 
the  corresponding  probabilities — as  is  proven  by  the  contro- 
versies concerning  the  determination  of  true  probabilities 
of  death — can  only  with  difficulty  be  obtained  in  a  mathe- 
matically correct  way. 

But  even  if  the  series  in  question  consists  of  numerical 
probabilities  the  application  of  the  theory  of  probability  is, 
nevertheless,  not  without  difficulty.  Lexis  and  von  Bortkie- 
wicz  assert  that  relative  numbers  can  only  be  considered 
as  empirical  probabilities  if  they  belong  to  a  series  of 
values  which  are  grouped  around  their  mean  according  to 
the  theory  of  probability.  Given  such  a  series,  which  is 
called  "  typical  "  because  the  mean  is  typical,  then  it  is 
natural  to  imagine  that  a  theoretical  probability  exists 
which  appears  with  merely  accidental  errors  in  the  items 
of  the  series,  which  items  usually  refer  to  different  geo- 
graphical districts  or  periods  of  time.  Now,  we  are  en- 
titled to  compute  the  precision  of  the  items  as  well  as  the 
precision  of  the  mean.  At  all  events  we  can  assume  that 
the  latter  is  a  closer  approximation  to  the  theoretical  prob- 
ability than  any  one  of  the  items.  However,  series  of  rela- 
tive numbers  which  not  only  obey  certain  formal  conditions, 
but  also  show  dispersion  around  a  typical  mean  according 

**  What  relative  numbers  these  are  has  been  explained  above,  pp. 
19  flf. 


THE  ARITHMETIC  MEAN  177 

to  the  theory  of  probability,  are  very  rare  in  statistics.*' 
IMost  statistical  series  do  not  show  a  sufficient  coincidence 
with  the  theory  of  probability,  so  that  that  theory  can 
only  be  applied  in  comparatively  rare  cases  to  the  mathe- 
matical proof  of  the  superiority  of  the  arithmetic  mean. 
The  computation  of  the  arithmetic  mean  from  a  series  of 
relative  numbers  can  only  in  very  rare  cases  be  justified 
by  the  assumption  that  the  arithmetic  mean  is  more  valu- 
able scientifically  from  the  standpoint  of  the  theory  of 
probability  than  the  individual  relative  numbers.  Often 
the  use  of  the  mean  arises  because  of  practical  reasons  which 
call  for  a  simplified  result.  Often  the  use  of  the  mean  is 
even  undesirable,  since  characteristic  differences  which  in- 
dividual constituents  exhibit  are  thus  frequently  oblit- 
erated. 

The  use  of  the  calculus  of  probability  for  the  determina- 
tion of  the  degree  of  accuracy  of  a  statistical  relative 
number  conceived  as  an  empirical  probabilit}-  has  been, 
therefore,  to  a  great  extent  relegated  to  the  background 
by  the  modem  mathematical  statisticians,  especially  by 
Lexis,  while  the  older  theoretical  statisticians  laid  great 
stress  on  it.  Lexis  thinks  that  the  chief  advantages  of 
the  calculus  of  probability  as  applied  to  statistics  are  that 
it  offers,  first,  a  comprehensive  scheme  for  frequency  dis- 
tributions and,  second,  a  measure  of  the  stability  of  statis- 
tical relative  numbers.*^"*^ 


*°  This  holds  for  series  with  "  normal "  as  •well  as  for  series  with 
"  supra-normal "  dispersion,  of  which  the  former — according  to  the 
explanation  of  Lexis — correspond  to  a  constant  probability,  the  lat- 
ter to  a  probability  subject  to  accidental  fluctuations  (cf.  below, 
Part  III,  Chap.  IV,  C). 

*' Cf.  V.  Bortkiewicz,  "Die  Theorie  der  Bcvolkerungs-  und  Moral- 
statistik  nach  Lexis,"  Conrad's  Jahrbiicher,  3rd  series,  Vol.  XXVII 
(1904),   p.   247. 

*''  The  questions  connected  with  those  problems  of  the  theory  of 
probability  are  treated  in  the  third  part  of  the  book,  Chap.  IV,  C. 


178  THE  VARIOUS  KINDS  OF  AVERAGES 

3.  RELATION  OP  * '  MATHEMATICAL  "  TO  "  NON-MATHEMATICAL ' ' 
STATISTICS ;  TYPICAL  AND  ATYPICAL  ARITHMETIC  MEANS 

The  theorems  of  the  theories  of  errors  and  of  probability 
constitute  a  mathematically  precise  statement  of  ideas  which 
are  familiar  to  non-mathematical  statisticians  and  even  to 
laymen  lacking  any  knowledge  of  statistics.  It  is  a  matter 
of  common  knowledge  that  measurements  of  an  object  are 
usually  affected  with  accidental  errors;  that  an  average 
from  several  measurements  gives  the  true  size  of  an  object 
most  correctly;  and  that  the  accuracy  of  the  average 
increases  with  the  number  of  measurements.  It  is  obvious 
to  every  statistician  that  a  single  statistical  value  may  differ 
greatly,  on  account  of  individual  causes,  from  the  "  nor- 
mal "  quantity  desired  and  that  by  computing  an  average 
from  a  greater  number  of  homogeneous  values  disturbing 
causes  are  eliminated  and  a  more  reliable  foundation  is 
obtained.*^     It  is  likewise  obvious  that  relative  numbers 

**  This  knowledge  is  the  reason  that,  for  various  practical  purposes, 
we  are  not  satisfied  with  the  number  for  a  single  year,  as  this 
number  may  be  influenced  by  exceptional  circumstances,  but  try  to 
support  our  conclusions  by  the  average  results  of  several  years. 
Thus,  most  of  the  mortality  tables  were  not  computed  on  the  basis 
of  the  results  of  the  census  and  the  deaths  of  one  year,  but  on 
the  basis  of  the  average  of  several  censuses  and  of  the  average 
deaths  of  several  years.  In  the  computation  of  the  German  mortality 
table  of  the  year  1887,  the  results  of  the  census  of  each  of  the 
years  1871,  1875  and  1880  and  the  death  rates  of  the  period  1871-1881 
were  used.  That  the  conditions  of  a  single  year  cannot  always  be 
the  standard  is  also  taken  into  consideration  in  the  different  fields  of 
legislation.  Thus  §  59  of  the  Austrian  Public  School  law  decrees  that 
schools  must  be  built  wherever  the  number  of  school  children  that 
have  to  go  to  a  school  more  than  one  mile  distant  averages  above 
40  for  a  period  of  5  years.  On  the  occasion  of  the  Bosnian  tax- 
census  in  the  year  1905  it  was  decreed  that  the  taxes  of  every  com- 
munity should  be  the  average  amount  of  the  contributions  of  the 
community  during  the  last  10  years.  Many  other  examples  could 
be  given. 


THE  ARITHMETIC  MEAN  179 

which  are  based  on  small  numbers  of  observations  are 
unreliable  and  that  the  accuracy  of  a  ratio  generally  in- 
creases with  the  size  of  the  quantities  brought  into  relation. 
However,  this  generally  recognized  connection  between 
the  number  of  observations  and  the  precision  of  the  average 
or  the  statistical  probability  cannot  be  stated  exactly  with- 
out using  mathematical  methods.  The  important  question, 
When  is  a  statistical  quantity  large  enough  to  allow  a  con- 
clusion? has  been  frequently  raised  by  non-mathematical 
statisticians.  No  satisfactory  general  answer  to  this  ques- 
tion has  been  given.  As  a  method  to  enable  us,  in  an 
individual  case,  to  answer  the  question  of  the  legitimacy 
of  a  conclusion  with  regard  to  the  size  of  a  quantity  in 
question,  the  division  of  the  observations  into  several  sets 
has  often  been  proposed.  If  these  constituent  sets  result 
in  values  not  greatly  divergent,  then  the  totality  is  great 
enough  so  that  the  law  of  great  numbers  operates,  other- 
wise the  totality  is  not  great  enough.  According  to  the 
principles  of  the  theory  of  errors  or  the  theory  of  proba- 
bility there  is  no  certain  limit  when  a  totality  may  be 
considered  to  be  great  enough  in  general.  The  greater 
the  totality,  the  greater  the  precision  of  the  average  or  the 
probability  computed  for  it.  The  constituents  resulting 
from  the  division  of  a  totality  obviously  exhibit  divergent 
values,  since  they  comprise  fewer  observations,  and  the  aver- 
ages and  probabilities  of  the  constituents  fluctuate  within 
definite  limits  around  the  average  or  the  probability  of 
the  totality,  these  limits  being  rather  far  apart  if  the 
numbers  of  observations  are  small.  Some  non-mathemat- 
ical statisticians  think  that  the  existence  of  sufficiently 
large  quantities  is  proven  if  a  series  of  values  shows  a 
regular  formation — for  instance,  if  a  series  of  death  proba- 
bilities increases  or  decreases  in  definite  proportion  with 
the  age.  But  this  regularity  is  never  perfect  and  its  de- 
gree also  depends  on  the  size  of  the  quantities  on  which 
the  items  are  based. 


180  THE  VARIOUS  KINDS  OF  AVERAGES 

This  discussion  proves  sufficiently  the  importance  of  the 
mathematical  method  for  certain  problems  of  statistics. 
On  this  question  von  Bortkiewicz  expresses  himself  as  fol- 
lows: "It  is  a  self-delusion  to  believe  that  we  are  able 
to  work  independently  of  the  ideas  of  the  theory  of  proba- 
bility. In  reality  even  the  grimmest  foe  of  the  analogy  with 
chance  is  ruled  in  his  statistical  work  by  conceptions  that 
originate  from  that  theory.  Indeed,  the  scientific  statis- 
tician daily  asks  himself  the  question,  if  in  this  or  that 
case  the  amount  of  the  material  at  hand  is  sufficient  for 
accidents  to  neutralize  or  counterbalance  each  other.  He  ap- 
plies, therefore,  the  theory  of  probability  without  his  own 
will  and  knowledge  and,  consequently,  in  an  unmethodo- 
logical  way,  in  the  rough  way  of  the  pure  empiricist."*'' 
Indeed,  there  is  a  danger  in  the  fact  that  non-mathematical 
statisticians  sometimes  unconsciously  apply  certain  theo- 
rems of  the  theories  of  errors  and  probability  when  their 
use  is  not  warranted.  In  this  class  belongs  principally 
the  blind  worship  for  great  numbers — widely  spread,  espe- 
cially in  former  times — and  the  desire  originating  from 
this  feeling  to  combine  great  numbers  of  observations  in 
order  to  compute  an  average  or  to  obtain  a  relative  number. 
If  we  consider  the  values  combined  in  the  light  of  the 
theories  of  errors  and  probability,  then  we  find  in  many 
cases  that  with  reference  to  the  series  of  items  we  cannot 
assume  at  all  that  the  value  of  the  average  or  the  relative 
number  increases  with  the  number  of  observations.  The 
material  with  which  practical  statistics  has  to  work  really 
offers  few  opportunities  for  the  direct  application  of 
the  principles  and  methods  of  the  theories  of  errors  and 
probability.  But  the  comparison  of  statistical  material 
with  these  theories  always  offers  an  interesting  standard  for 
judging  this  material,  and  therefore  is  useful  even  if  it 
occasionally  results  merely  in  the  negative  fact  that  certain 

"  "  Die  Theorie  der  Bevolkerungs-  und  Moralstatistik  nach  Lexis," 
Conrad's  Jahrbiicher,  3rd  Series,  Vol.  XXVII  (1904),  p.  251  f. 


THE  ARITHMETIC  MEAN  181 

theorems  of  the  theories  of  errors  and  probability  some- 
times used  intuitively  by  non-mathematical  statisticians 
cannot  be  applied  to  the  case  at  hand. 

The  differentiation  between  typical  and  non-typical  series 
and  means,  made  in  mathematical  as  well  as  non-mathe- 
matical statistics,  furnishes  an  especially  interesting  ex- 
ample of  the  common  use  of  ideas  drawn  from  the  theories 
of  error  and  probability.  Mathematical  statistics  sees  in 
a  typical  series  one  corresponding  to  the  normal  law  of 
error  or  the  theory  of  probability,  a  series  "  whose  items 
are  approximations  affected  by  accidental  errors,  to  a  fixed 
base  value."  ^°  A  typical  series  naturally  results  in  a 
typical  mean,  a  non-typical  series  in  a  non-typical  mean. 
In  non-mathematical  statistics,  likewise,  series  and  espe- 
cially arithmetic  means  are  divided  into  typical  and  non- 
typical  groups  (Bertillon,  Block,  Haushofer,  etc.).  Haus- 
hofer  says:  **  The  average  may  be  a  value  which  is  ap- 
proached by  all  the  phenomena  observed,  with  which  they 
sometimes  are  almost  identical.  Then  it  is  called  a  type  of 
all  the  single  phenomena."  "  But,  on  the  other  hand, 
the  average  may  be  merely  an  arithmetic  abstraction,  i.  e., 
a  value  which,  although  having  been  computed  from  the 
members  of  a  series,  is  not  intimately  connected  with  the 
single  items.  The  average  age  of  the  population  is  an 
example.  No  definite  line  can  be  drawn  between  these  two 
kinds  of  averages.  The  greater  the  differences  of  the  single 
phenomena  from  which  the  averages  have  been  computed, 
the  closer  the  latter  approach  to  mere  arithmetic  abstrac- 
tions."^^ Series  are  classified  in  non-mathematical  sta- 
tistics as  typical  or  non-typical,  depending  upon  the  arith- 
metic mean  resulting  from  them,  whether  typical  or  non- 
typical. 

"°  Lexis,  Abhandlimgen  zur  Theorie  der  Bevolkerungs-  und  ^Moral- 
statistik,  VIII,  "On  the  Theory  of  Stability  of  Statistical  Series," 
p.  171. 

"  Lehr-  und  Handbuch  der  Statistik,  2nd  ed.,  p.  53  f. 


182  THE  VARIOUS  KINDS  OF  AVERAGES 

Non-mathematical  statistics,  as  well  as  mathematical  sta- 
tistics, denotes,  as  typical  averages,  those  which  represent 
series  in  a  peculiarly  qualified  way,  and  the  series  thus 
represented  are  called  typical  series.  However,  while  math- 
ematical statistics  can  use  the  theories  of  errors  and  prob- 
ability as  a  measure  to  ascertain  typical  series  and  typical 
means  with  certainty,  a  similar  precise  and  objective  meas- 
ure is  lacking  in  non-mathematical  statistics,  and  the  de- 
cision whether  a  certain  mean  or  a  certain  series  may  be 
called  typical  becomes  more  or  less  subjective.  In  general, 
non-mathematical  statistics  sees  in  a  typical  mean  one 
which  corresponds  to  a  great  number  of  items,  and  around 
which  the  whole  series  is  distributed  as  symmetrically  as 
possible  and  without  very  great  deviations.  Means  not 
answering  these  conditions,  i.  e.,  means  which  lie  outside 
of  the  main  group  of  items  or  around  which  the  items  are 
not  distributed  symmetrically,  are  called  non-typical.  A 
definite  line,  however,  cannot  be  drawn  between  typical  and 
non-typical  means  in  elementary  statistics. 

Since  non-mathematical  statistics,  as  has  been  mentioned, 
are  lacking  the  objective  criteria  at  the  disposal  of  mathe- 
matical statisticians  for  the  differentiation  between  typical 
and  non-typical  series  and  means,  different  conclusions  may 
easily  result  in  individual  cases.  A  series  of  relative  num- 
bers and  their  mean  may  seem  to  be  typical  to  the  non- 
mathematical  statistician,  while  the  mathematician  may  not 
be  able  to  find  correspondence  with  the  theory  of  probabil- 
ity; a  series  of  measurements  may  appear  to  the  former 
to  be  distributed  with  sufficient  symmetry  to  denote  a  typ- 
ical series  and  a  typical  mean,  while  the  latter  may  find  a 
contradiction  to  the  Gaussian  law.  In  general,  the  rules 
of  the  non-mathematical  statisticians  are  less  strict;  this 
enables  them  to  use  the  expression  "  typical  "  to  a  con- 
siderable extent,  while  the  mathematical  statisticians,  as 
is  known,  have  found  but  few  series  which  might  be  called 
*'  typical  "  in  their  sense  of  the  word. 


THE  ARITHMETIC  MEAN  183 

But  even  if  the  idea  of  the  "  typical  average  "  is  defined 
in  a  manner  not  too  rigorous,  as  is  customary  in  non- 
mathematical  statistics,  even  then  decidedly  non-typical 
averages  occur  only  too  frequently.  These  "  mere  arith- 
metic abstractions  "  can  be  used  to  represent  a  series  only 
with  great  precaution.  The  best  known  example  of  such 
a  non-typical  average — besides  the  average  age  of  the  living 
obtained  from  the  census  figures — is  the  expectation  of 
life,  the  arithmetic  mean  of  the  ages  in  the  mortality  table. 
In  Prussia  the  mean  expectation  of  life  for  the  period  1881- 
1890  was  39  years,  one  month,  in  Austria  33  years  and 
8  months.  In  all  countries  it  falls  in  the  middle  age  classes 
which  show  relatively  few  deaths,  while  the  majority  of 
deaths  occur  in  childhood  and  old  age.^^ 

"  G.  V.  Mayr'g  use  of  the  terms  "  typical "  and  "  non-typical " 
means  differs  from  the  prevailing  usage  (Theoretische  Statistik,  p. 
102).  V.  Mayr  says:  "A  typical  mean  is  given,  if  from  the  nature 
of  the  thing  an  ascertained  average  also  represents  a  possible  reality 
of  the  phenomena  in  question.  This  is  the  case  with  birth,  death, 
and  crime  rates  in  time  series,  in  absolute  as  well  as  in  relative 
nimibers.  A  merely  arithmetic  abstraction  is  given  if  an  actual 
coincidence  of  all  the  objects  with  the  average  cannot  reasonably 
be  conceived.  Such  is  the  case  with  the  ascertained  average  age  of 
those  living  and  those  dying."  A  typical  mean  in  v.  Mayr's  sense 
is  the  average  of  those  marrying  in  contrast  to  the  average  age  of 
those  living,  v.  Mayr  says  in  this  connection  (Boviilkerungsstatistik, 
p.  402):  "The  average  age  of  the  living  population  is  merely  an 
arithmetic  abstraction;  the  existence  of  a  population  consisting  only 
of  people  of  average  age  is  not  conceivable.  But  it  is  not  incon- 
ceivable that  all  those  marrying  do  so  always  at  the  same  age." 

G.  V,  Mayr  has  also  appropriated  the  term  "  typical  series,"  con- 
trary to  the  general  statistical  terminologj%  for  a  peculiar  category  of 
series  which  he  has  formed.  He  says  (Theoretische  Statistik,  p.  90)  : 
"  Typical  series,  in  opposition  to  those  series  which  offer  only  a  con- 
crete section  of  a  continuous  development,  are  those  which  from 
the  nature  of  the  observed  material  represent  within  themselves  all 
the  possibilities  of  a  given  phenomenon  (for  instance,  distribution  of 
a  given  population  according  to  height,  of  a  given  number  of 
births  according  to  the  number  of  children  born  in  one  confinement, 


184  THE  VARIOUS  KINDS  OF  AVERAGES 

A  remarkable  change  has  occurred  in  the  use  of  the  word 
typical.  By  typical  phenomena  the  earlier  writers  meant 
processes  which  are  governed  by  certain  laws  of  nature — 
for  instance,  most  of  the  physical  and  chemical  processes. 
It  had  been  found  that  with  such  "  typical  "  phenomena 

and  according  to  the  sex  of  the  children,  distribution  of  births, 
deaths,  crimes  over  the  diflferent  seasons,  etc.).  With  such  typical 
series  a  natural  arrangement  of  the  members  results  from  the 
quantitative  or  time  graduation  of  the  possibilities  of  the  phe- 
nomenon." In  the  course  of  his  discussion,  however,  v.  Mayr  ap- 
proaches the  conception  defined  by  the  mathematical  statisticians 
and  adopted  also  by  the  elementary  statisticians  in  general.  For 
V.  Mayr  continues :  "  The  most  pronounced  form  of  such  typical 
series  seem  to  be  those  in  which  the  items  of  a  concrete  series  of 
observations  must  be  considered  to  be  quasi-inaccurate  representa- 
tions of  an  unchanging  base-value,  which  is  expressed  in  the  phe- 
nomena actually  observed  only  with  purely  accidental  errors.  The 
symmetrical  arrangement  of  the  items  located  below  and  above  the 
mean  characterizes  this  most  pronounced  form  of  typical  series  (fre- 
quency curves)  ;  the  more  asymmetrical  this  arrangement  and  the 
less  decided  a  central  elevation  of  the  curve,  the  more  does  the  series 
lose  its  typical  character." 

Adolphe  and  Jacques  Bertillon,  as  well  as  Block,  call  non-typical 
means  "  moyennes  indices "  in  opposition  to  "  moyennes  typiques," 
and  the  former  expression  also  denotes  certain  isolated  averages 
for  potential  measurements.  J.  Bertillon's  principal  examples 
for  "  moyennes  indices "  are  the  mean  expectation  of  life  and  the 
average  consumption  of  alcohol  per  capita  of  the  population  (cf. 
A.  Bertillon,  "  La  thgorie  des  moyennes  en  Statistique,"  Journal  de  la 
Soci6t6  de  Statistique  de  Paris,  1876,  p.  268,  and  J.  Bertillon,  Cours 
gl^mentaire  de  Statistique,  p.  118  f.,  and  Block,  Traits  th^orique  et 
pratique  de  Statistique,  2nd  ed.,  p.  124).  The  astronomer  Herschel, 
who  wrote  the  preface  to  Quetelet's  Physique  sociale,  proposed  to 
denote  non-typical  means  also  in  France  by  the  English  word  aver- 
age; but  this  word  has  not,  in  English,  the  meaning  which  Herschel 
attributes  to  it  and  it  has  found  no  place  in  the  French  language. 
Quetelet  himself  proposed  to  call  typical  means  simply  "  moyennes," 
and  to  use  the  name  "  moyenne  arithmetique  "  for  non-typical  means. 
This  terminology  was  not  accepted.  A.  Bertillon  objected  for  the 
reason  that  it  had  not  been  chosen  properly,  since  typical  means 
are  also  computed  in  an  arithmetical  way. 


THE  ARITHMETIC  MEAN  185 

the  law  established  by  the  observation  of  a  single  case  ought 
to  hold  without  exception  for  all  analogous  cases.  If  a 
physicist  has  observed  that  a  drop  of  mercury  freezes 
at  a  certain  temperature,  then  he  can  assume  that  this 
fact  holds  for  all  the  drops  of  mercury  in  the  world." 
The  older  statisticians  used  to  emphasize  that  such  "  typ- 
ical "  phenomena  are  not  suitable  objects  for  statistical 
research  and  that  it  is  the  province  of  statistics  to  investi- 
gate "  individual  "  phenomena  (i.  e.,  those  that  vary  from 
individual  to  individual),  or  as  Meitzen  ^*  says:  "  To  strive 
for  the  perception  of  the  '  non-typical,'  attainable  only 
through  the  statistical  method." 

Furthermore,  the  older  statisticians  often  made  the  mis- 
take of  confusing  the  antithesis  between  "  typical  "  and 
"  non-typical  "  phenomena  with  that  between  nature  and 
society,  because  generally  they  considered  the  phenomena 
of  nature  to  be  "  typical  "  and  the  phenomena  of  society 
to  be  "non-typical."^^  As  a  matter  of  fact,  however, 
these  distinctions  do  not  coincide.  It  is  true  that  social 
phenomena,  as  a  rule,  are  non-typical.^^  They  are  not  gov- 
erned by  one  but  by  various  causes  of  different  importance, 
so  that  every  phenomenon  shows  a  more  or  less  individual 
aspect.  We  cannot  draw  any  conclusions  as  to  the  dura- 
tion of  life,  the  age  of  marriage,  or  income  of  specified  in- 

"  Cf.  Haushofer,  Lehr-  u.  Handbuch  der  Statistik,  2nd  ed.,  p.  38. 

'*  Geschichte,  Theorie,  imd  Technik  der  Statistik,  2nd  ed.,  p.  81. 

"'  Cf.,  for  instance,  Riimelin,  "  Zur  Theorie  der  Statistik,"  I 
(Reden  und  Aufsiitze,  1875,  p.  213  ff.). 

"  However,  this  rule  is  not  without  exceptions.  In  social  life,  and 
especially  in  economic  life,  there  exist  typical  processes,  which  orig- 
inate from  a  definite  motive  and,  given  the  same  conditions,  always 
repeat  themselves.  Lexis  (Theorie  der  Massenerscheinungen,  pp.  2-4), 
in  such  cases,  speaks  of  generic  quantitative  phenomena  and  gives 
the  following  example :  If  on  the  Berlin  exchange  the  exchange  rate 
on  Paris  goes  above  81.40,  then  it  may  be  asserted  that  all  German 
bankers,  who  are  at  all  prepared  for  operations  of  arbitrage,  will 
send  gold  to  Paris. 


186  THE  VARIOUS  KINDS  OF  AVERAGES 

dividuals  from  the  knowledge  of  those  facts  for  other 
individuals.  However,  besides  the  "  typical  "  phenomena, 
i.  e.,  processes  which  according  to  the  older  terminology 
are  governed  by  laws  of  nature  with  which  statistics  has 
nothing  to  do,  nature  also  exhibits  "  non-typical,"  "  in- 
dividual "  phenomena  in  great  abundance,  a  fact  which 
older  statisticians  do  not  seem  to  have  observed  sufficiently. 
Thus,  dimensions  of  animals  and  plants  are  extremely 
variable,  and  therefore  quantitative  single  observations  are 
indispensable  in  biological  research.  Furthermore,  meteor- 
ological phenomena,  especially,  are  very  changeable  and 
consequently  require  quasi-statistical  quantitative  observa- 
tions. 

In  modern  statistics,  the  term  "  typical  "  does  not  serve 
to  distinguish  different  categories  of  phenomena  but  prin- 
cipally to  indicate  definite  series  and  means.  The  observa- 
tion of  those  phenomena  which  the  older  statisticians  called 
**  non-typical  "  on  account  of  their  individual  differences, 
results  in  statistical  series  and  means  which  the  modern 
statisticians  divide  into  "  typical  "  and  '*  non-typical  " 
according  to  the  distribution  of  the  items  around  the  mean 
(thus  following  a  criterion  which  is  completely  different 
from  the  criterion  on  which  the  older  distinction  between 
"  typical  "  and  "  non-typical  "  was  based). 


4.  MATHEMATICAL  METHODS  OP  JUDGING  THE  SIGNIFICANCE 
OF  THE  DIFFERENCE  BETWEEN  TWO  ARITHMETIC  MEANS 
OR  STATISTICAL  PROBABILITIES 

The  principles  of  the  theories  of  error  and  probability 
are  frequently  used  by  mathematical  statisticians  when 
comparing  statistical  averages  or  relative  numbers  which 
have  the  form  of  numerical  probabilities.  From  the  differ- 
ence of  two  means  or  relative  numbers,  as  shown  in  a  pre- 
vious chapter,  we  may,  under  certain  circumstances,  draw 


THE  ARITHMETIC  MEAN  187 

inferences  concerning  the  general  causes  acting  on  the  phe- 
nomena compared."  While  doing  this  we  must  distinguish 
between  the  comparison  of  quantities  differing  in  geograph- 
ical location  and  time,  and  those  differing  quantitatively  or 
qualitatively.  By  comparing  values  which  refer  to  different 
geographical  or  time  divisions,  we  can  ascertain  the  ex- 
istence of  differences,  but  we  do  not  thereby  determine 
the  causes  acting  on  the  quantities  in  question.  If  we 
find,  e.  g.,  that  the  death  rates  of  two  countries  differ,  we 
can  infer  that  the  causal  conditions  influencing  the  mor- 
tality differ  in  the  two  countries,  but  for  the  time  being 
we  do  not  know  wherein  this  difference  lies.  But  if  we 
compare  masses  differing  quantitatively  or  qualitatively 
in  a  definite  way  and  find  a  difference  between  the  means 
or  relative  numbers  ascertained  for  the  masses  in  ques- 
tion, then  we  can,  under  defined  conditions,  ascertain  im- 
mediately the  cause  of  this  difference.  For  if  certain 
postulates  are  fulfilled  we  can  then  trace  the  difference  of 
the  means  or  relative  numbers  at  hand  back  to  the  quanti- 
tative or  qualitative  difference  between  the  masses  on 
which  these  values  are  based.  If  we  find  that  the  death 
rate  of  males  is  higher  than  that  of  females,  or  that  the 
death  rates  of  those  belonging  to  different  occupations  differ 
from  each  other,  then  we  are  justified,  other  things  being 
equal,  in  attributing  to  the  sex  or  the  occupation  a  decisive 
influence  on  the  mortality. 

However,  we  can  infer  that  different  fundamental  causes 
affect  the  items  (having  space,  time,  quantitative  or  quali- 
tative differences)  on  which  the  compared  means  or  relative 
numbers  are  based,  only  in  case  the  difference  between 
the  compared  values  is  significant.  No  inference  can 
be  drawn  from  very  small  differences.  "  One  does  not 
need  a  scientific  training  in  order  to  be  aware  of  this.  It 
is  evident  that  it  is  not  a  sign  of  improvement  of  sanitary 
conditions  if  in  a  certain  locality  12,345  deaths  occur  in 

*'  Cf.  p.  110  S. 


188  THE  VARIOUS  KINDS  OF  AVERAGES 

one  year  and  12,344  deaths  in  the  next  year.  The  differ- 
ence is  so  small  that  it  may  be  accidental.  But  what  does 
accidental  mean  in  this  connection?  What  are  its  limita- 
tions? What  differences  are  great  enough  to  pass  from 
the  realm  of  the  accidental  ?  If  the  answers  to  these  ques- 
tions are  to  be  universal  and  not  merely  depend  upon  the 
idiosyncrasy  of  the  statistician,  which  may  lead  the  skeptical 
to  an  overestimation  of  the  limits  of  the  accidental  and 
the  sanguine  to  rash  conclusions,  then  they  must  be  based 
on  the  law  of  great  numbers. ' '  ^* 

As  a  matter  of  fact,  mathematical  statisticians  have  de- 
veloped special  methods  that  enable  them  to  ascertain  the 
significance  or  lack  of  significance  from  the  standpoint  of 
the  theories  of  error  and  probability  of  the  numerical 
difference  between  two  arithmetic  means  (means  for  an 
element  of  measurement),  or  between  two  relative  num- 
bers that  may  be  assumed  to  represent  empirical  proba- 
bilities. 

Mathematical  statisticians  consider  means  computed  from 
elements  of  measurement  to  be  mere  approximations  to  the 
proper  theoretical  normal  value  of  the  quantity  in  question. 
This  normal  value  is  reflected  in  the  concrete  mean,  but 
it  is  not  expressed  by  it  with  perfect  accuracy  on  account 
of  the  limited  number  of  cases  on  which  the  mean  is  based. 
Therefore,  different  empirical  means  may  correspond  to  the 
same  theoretical  normal  value  and,  on  the  other  hand,  the 
same  empirical  mean  may  be  based  on  different  theoretical 

*'  A.  A.  Tschuprow,  Die  Aufgaben  der  Theorie  der  Statistik,  Jahr- 
bucher  f iir,  etc.,  edited  by  Gustav  Schmoller,  29th  year  ( 1905 ) ,  2nd 
number,  p.  36.  An  example  of  judging  a  difference  with  and  without 
the  application  of  the  theory  of  probability,  is  found  in  Edgeworth, 
"  Methods  of  Statistics,"  Jubilee  Volume  of  the  Roy.  Stat.  Soc,  p.  206. 
There  Edgeworth  asserts  that  in  a  certain  case  where  Wappaus,  in 
comparing  data  for  two  periods  of  time,  had  assumed  a  change  in 
the  mortality,  a  closer  investigation  by  means  of  the  theory  of  proba- 
bility shows  no  sufficient  reason  for  assuming  a  change  other  than 
accidental  fluctuations. 


THE  ARITHMETIC  MEAN  189 

normal  values.  Since,  according  to  the  opinion  of  mathe- 
matical statisticians,  only  the  theoretical  normal  values  (the 
accidental  errors  being  eliminated)  are  determined  by  the 
general  causes,  therefore  a  difference  in  the  fundamental 
causes  operating  upon  two  quantities  can  be  inferred  only 
in  case  a  difference  between  the  theoretical  normal  values 
of  these  quantities  can  be  proven.  Therefore  the  point  in 
question  is  to  ascertain  if  (and  with  what  probability)  the 
concrete  empirical  values,  in  spite  of  their  numerical  differ- 
ence, are  to  be  taken  as  approximations  to  the  same  theoret- 
ical normal  value,  or  if,  on  account  of  this  difference,  they 
must  be  considered  as  approximations  to  different  normal 
values.  If  it  is  very  probable  that  the  compared  means  are 
approximations  to  the  same  theoretical  normal  value,  then 
the  difference  between  the  means  is  insignificant ;  if, 
however,  the  probability  that  the  two  means  represent 
the  same  normal  value,  is  small,  then  the  difference  is 
significant  and  an  inference  of  different  causation  is  allow- 
able. 

The  practical  application  of  the  above  theorems  can  be 
illustrated,  without  reproduction  of  mathematical  details, 
by  an  example  taken  from  an  article  by  Professor  Edge- 
worth.  He  compares  the  average  height  of  2,315  criminals 
with  the  average  height  of  8,585  persons  of  the  normal 
population ;  the  former  average  is  2  inches  lower  than 
the  latter.  Under  the  supposition  that  criminals  have,  in 
general,  the  same  heights  as  the  whole  population,  a  modulus 
of  0.08  inch  is  found  for  the  difference  of  two  averages 
with  the  numbers  of  observation  mentioned.  If  the  actual 
difference  between  the  two  averages  compared  were  not 
more  than  three  times  this  modulus,  then  we  could  assume 
that  this  difference  is  merely  accidental,  i.  e.,  caused  merely 
by  the  small  number  of  observations  and  by  the  accidental 
errors  attached  to  these.  But  the  actual  difference  (2 
inches)  is  much  larger  than  three  times  the  modulus 
(0.24  inch)  ;  therefore,   the  difference  is  significant  and 


190  THE  VARIOUS  KINDS  OF  AVERAGES 

permits  the  inference  that  criminals  are,  on  the  average, 
of  shorter  stature  than  the  normal  population.^^ 

This  method  of  comparing  averages  may  be  used  espe- 
cially to  ascertain  periodical  fluctuations.  The  differences 
existing  between  certain  months  of  a  single  year  are  of 
course  not  conclusive ;  monthly  averages  must  be  computed 
and  compared  for  a  number  of  years.  The  method  may 
also  be  applied  to  investigate  whether  or  not  a  series  shows 
a  certain  direction  of  development.  For  this  purpose  the 
averages  resulting  from  successive  periods  are  compared  to 
ascertain  if  the  differences  between  them  must  be  con- 
sidered significant  or  if  they  are  merely  accidental.®" 

The  method  of  comparing  statistical  probabilities  is  based 
on  considerations  similar  to  those  used  above  in  the  com- 
parison of  arithmetic  means  for  an  element  of  measurement. 
The  probability  is  ascertained  with  which  the  two  numbers 
compared  may  be  considered,  with  regard  to  the  numerical 
difference  existing  between  them,  to  be  empirical  values 
of  the  same  theoretical  probability.  If  this  probability  is 
great,  then  the  difference  between  the  two  numbers  is  in- 
significant and  must  be  due  to  accidental  causes.  If,  how- 
ever, there  is  but  slight  probability  that  the  numbers  com- 
pared are  empirical  values  of  the  same  theoretical  proba- 
bility, then  the  difference  between  these  numbers  is  sig- 
nificant, and  it  must  be  assumed  that  the  phenomena  com- 
pared reflect  different  theoretical  probabilities  and  it  fol- 


"  Cf.  "  Methods  of  Statistics,"  Jubilee  Volume  of  the  Royal 
Statistical  Society  (pp.  187  f.  and  195  f.),  where  numerous  other 
examples  of  the  method  under  discussion  are  given.  A  mathematical 
method  similar  to  Edgeworth's  is  used  by  Westergaard  (Grundzuge 
der  Theorie  der  Statistik,  p.  187),  who  evaluates  the  actual  diflference 
by  using  the  mean  error  of  the  difference  of  the  two  averages. 
See  also  the  discussions  by  v.  Bortkiewicz,  "  Kritische  Betrachtungen 
zur  Theoretischen  Statistik,"  Conrad's  Jahrb.,  3rd  series,  Vol.  X 
(1895),  a,  p.  334-341. 

••  See  Bowley,  Elements  of  Statistics,  2nd  ed.,  p.  313  ff. 


THE  ARITHMETIC  MEAN  191 

lows  that  the  general  conditions,  which  determine  the  theo- 
retical probabilities,  are  different. 

An  example  of  the  practical  application  of  these  ideas 
is  found  in  the  paper  of  A.  A,  Tschuprow,  "  Die  Aufgaben 
der  Theorie  der  Statistik  "  (p.  37  f.).  Of  1,558,129 
inhabitants  33,181  died  in  Vienna  in  1897 ;  therefore 
the  empirical  mortality  amounts  to  21Voo ;  the  modulus 
equals  0.17Voo;  therefore  we  may  assume  that  the 
theoretical  probability  of  dying  is  within  the  limits  of 
2I.5IV00  and  20.49%o.  In  Prague,  in  the  same  year,  6,392 
persons  died  out  of  a  population  of  193,097;  the  mortality 
is  3IV00,  the  modulus  0.56Voo?  the  probability  of  dying, 
consequently,  must  lie  within  the  limits  of  29.32°/oo  and 
32.68  Voo-  Now  since  the  inferior  limit  for  Prague 
(29.32Voo)  is  considerably  higher  than  the  superior  limit 
for  Vienna  (21.51°/oo)  it  may  be  asserted  that  the  two  prob- 
abilities are  different,  and  that  the  conditions  of  life  with 
reference  to  the  mortality  are  actually  more  unfavorable 
in  Prague  than  in  Vienna.  The  real  cause  of  this, 
whether  it  is  bad  housing  conditions,  or  impure  drink- 
ing water,  or  unsanitary  occupations  of  the  population, 
or,  perhaps,  a  difference  of  the  sex  and  age  consti- 
tution of  the  population,  we  do  not  learn  for  the  time 
being  it  is  true,  but  by  ascertaining  that  the  conditions 
of  life  in  Vienna  and  Prague  with  reference  to  the  mor- 
tality are  not  the  same  we  gain  firm  ground  for  further 
research.^^ 

•^  For  the  evaluation  of  the  difference  between  two  empirical 
probabilities  various  other  methods  may  be  used,  for  instance  the 
method  of  comparing  the  actual  diflFerence  with  the  modulus  of  the 
diflference  of  frequencies  (Tschuprow,  loc.  cit.,  p.  38),  the  method 
of  comparing  the  actual  difference  with  the  mean  error  of  the  two 
frequencies  or  with  the  mean  error  of  the  difference  of  the  fre- 
quencies (Westergaard,  Die  Grundziige,  p.  45  f.  and  p.  81  f.,  and 
Die  Lehre  von  der  Mortalitiit  und  Morbilitiit,  p.  189  f.)  and  the 
method  of  comparing  the  actual  difference  with  the  probable  devia- 
tion computed   for  the   same    (Lexis,  Abhandlungen,   "The   Typical 


192  THE  VARIOUS  KINDS  OF  AVERAGES 

Besides  geographical  comparisons  we  can,  of  course,  also 
make  comparisons  of  different  time  periods,  classes  of  pop- 
ulation, etc.,  in  the  same  way.  We  may  investigate  whether 
a  phenomenon  shows  annual  fluctuations,  or  a  certain  direc- 
tion of  development,  etc. 

From  a  significant  difference  between  two  statistical  prob- 
ability figures  we  may  thus  infer  the  difference  of  the 
general  causes  acting  on  the  phenomena  compared.  It  is 
true  the  influence  of  the  ascertained  difference  of  causes 
cannot  be  determined  with  numerical  accuracy,  since  the 
difference  between  the  probability  figures  does  not  express 
it  clearly,  as  this  difference  is  also  influenced  simultaneously 
by  accidental  errors.  However,  it  is — as  Westergaard, 
particularly,  emphasizes — of  no  essential  importance  to  be 
able  to  express  numerically  the  difference  of  the  general 
causes;  the  main  object  is  simply  to  ascertain  that  such  a 
difference  of  the  general  causes  exists.®- 

The  method  of  the  mathematical  evaluation  of  the  differ- 

Values  and  the  Law  of  Error,"  p.  128).  See  also  Czuber  (Wahr- 
seheinliehkeitsreelmung,  No.  165,  "  Probability  That  Two  Empirical 
Determinations  are  Based  on  Unequal  Statistical  Probabilities,"  pp. 
304-307).  Cziiber  compares,  for  instance,  the  probability  of  a  male 
birth  among  legitimate  living  births  and  legitimate  still-births  and 
finds  that  it  can  be  asserted  almost  with  absolute  certainty,  that 
with  still-births  there  is  a  greater  probability  for  a  male  birth  than 
with  living  births. 

°^  It  is  true  that  a  difference  whose  effect  is  smaller  than  the 
limits  of  the  accidental  deviations  cannot  be  ascertained.  Wester- 
gaard (Die  Grundziige,  p.  58)  gives  the  following  example:  "If  the 
mortality  for  one  class  of  population  is  about  1%  higher  than  for 
the  population  in  general,  then  with  a  series  of  observations  of  100 
deaths,  with  a  mean  error  of  10,  we  will  never  be  able  to  assert  the 
influence  of  a  special  cause,  for  the  mean  error  is  many  times 
greater  than  the  average  effect  of  this  cause.  If  10,000  deaths  were 
given,  then  we  could  assume  such  a  cause  with  somewhat  greater 
accuracy,  but  its  effect  would,  on  the  average,  not  be  greater  than 
the  mean  error;  but  with  1,000,000  deaths  a  surplus  of  10,000  deaths 
would  betray  a  cause,  the  effect  of  which  would  surpass  the  mean 
error  ten  times  and,  therefore,  could  not  pass  as  accidental.     Here 


THE  ARITHMETIC  MEAN  193 

ence  between  two  relative  numbers  can  be  applied — as  can 
the  theory  of  probability  in  general — only  to  values  which 
formally  can  be  considered  to  be  probabilities  (or  functions 
of  probabilities).  This  means  a  considerable  limitation  of 
the  applicability  of  this  method  since  practical  statistics 
much  more  frequently  has  to  do  with  relative  numbers 
which  formally  are  not  probabilities,  than  with  relative 
numbers  which  fulfil  the  formal  conditions  in  question. 
Furthermore,  as  has  already  been  mentioned,  it  is  not  suf- 
ficient, according  to  the  opinion  of  modem  mathematical 
statisticians,  that  the  two  values  to  be  compared  are 
formal  probability  figures  if  taken  by  themselves.  It  must 
be  an  established  fact  that  these  values  belong  to  groups 
of  values  which  are  distributed  around  their  means  accord- 
ing to  the  theory  of  probability.  As  is  known,  these  con- 
ditions only  very  rarely  hold  true.  The  probabilities  of 
death,  for  instance,  compiled  for  a  number  of  years,  as  a 
rule  do  not  show  a  distribution  corresponding  to  the  theory 
of  probability.  Therefore,  strictly  speaking  it  is  not  legiti- 
mate to  apply  the  theory  of  probability  in  comparing  such 
items  as  probabilities  of  death  of  different  classes  of  the 
population.  Cases  are,  consequently,  very  rare  in  which 
the  application  of  the  theory  of  probability  to  the  measure- 
ment of  the  significance  of  the  difference  between  two 
relative  numbers  is  entirely  free  from  objection. 

again  the  importance  of  a  comprehensive  series  of  observations  is 
evident." 

The  above  limitation  also  holds  for  the  comparison  of  averages 
from  data  of  measurement.  Thus  it  cannot  be  ascertained  by  means 
of  the  theory  of  error,  for  instance,  whether  or  not  an  additional  duty, 
which  is  smaller  than  the  mean  error  of  the  price  fluctuations  of  the 
commodity  in  question,  has  had  any  effect  on  the  domestic  price. 


CHAPTER  III 

THE  GEOMETRIC  MEAN 

The  geometric  (or  logarithmic)  mean  of  n  items  is  the 
nth  root  of  their  product.  Where  the  items  are  represented 
by  Ej,  82  ...  an  the  formula  for  the  geometric  mean  is 

/ .        The  great  amount  of  arithmetic  work  in- 

y/  Ri  as &a 

volved  in  computing  the  geometric  mean  directly  is  lessened 
by  the  use  of  logarithms.  The  natural  number  correspond- 
ing to  the  arithmetic  mean  of  the  logarithms  of  a  number  of 
items  is  the  geometric  mean  of  those  items. 

The  geometric  mean  has  this  property  in  common  with 
the  arithmetic  mean,  that  in  its  computation  the  sizes  of  all 
the  items  are  of  decided  influence  on  the  size  of  the  mean.^* 
A  change  of  a  single  item  must  affect  the  numerical  size 
of  the  mean.  This  does  not  hold  for  the  median  or  the 
mode.  These  means  may  remain  unchanged  even  if  con- 
siderable parts  of  the  series  are  changed.  The  geometric 
mean  has  also  this  property  in  common  with  the  arithmetic 
mean,  that  it  may  not  coincide  with  any  of  the  items  used 
in  computing  it.  As  a  rule  the  geometric  mean  is  a  value 
which  does  not  occur  in  the  series  of  items.  If  items  of 
approximately  the  same  size  do  not  occur  at  all  or  only 

"  Tlierefore,  by  raising  the  geometric  mean  to  the  same  power 
as  there  are  items,  the  product  of  all  these  items  is  found,  just  as 
by  multiplying  the  arithmetic  mean  with  the  number  of  items  the 
sum  of  the  latter  is  obtained. 

'*  A  series,  in  which  an  item  equals  zero,  always  gives  zero  for  the 
geometric  mean,  without  regard  to  the  size  of  the  other  values 
of  the  series,  since  the  multiplication  of  any  number  by  zero  gives 
the  product  zero. 

194 


THE  GEOMETRIC  MEAN  195 

rarely,  we  may  call  the  geometric  mean  a  mere  arithmetic 
abstraction  and  we  must  consider  it  as  an  "  atypical  " 
mean. 

The  computation  of  the  geometric  mean  like  the  compu- 
tation of  the  simple  arithmetic  mean  presupposes  that  the 
items  have  equal  weights.  If,  in  a  concrete  case,  we  think 
that  this  supposition  does  not  correspond  to  fact,  then 
we  may  treat  this  case  by  a  method  similar  to  that  used 
in  the  computation  of  a  weighted  arithmetic  mean  from 
a  series  of  items  of  unequal  weights.  Before  computing 
the  geometric  mean  we  might  modify  the  series  by  raising 
every  item  to  the  power  which  indicates  its  importance  or 
weight  and  then  find  the  nth  root  of  the  product  of  the 
rectified  items,  where  n  equals  the  sum  total  of  the  weights. 
In  this  way  we  might  obtain,  so  to  speak,  a  weighted  geo- 
metric mean  of  the  series. 

The  geometric  mean  plays  a  very  subordinate  role  in 
practical  statistics.  It  has  been  used  by  statisticians  only 
sporadically,  for  instance,  by  Jevons  in  his  monograph  "  A 
Serious  Fall  in  the  Value  of  Gold  "  (1863)  for  the  com- 
putation of  the  mean  index  number  from  the  single  indices 
indicating  the  price  fluctuation  of  various  commodities.®^ 

The  geometric  mean  is  never  greater  than  the  arithmetic 
mean  of  a  series  of  items.®^*  The  difference,  however,  is  usu- 

"  See  also  the  article  "  On  the  Variation  of  Prices,"  etc.,  by 
Jevons  in  the  Journ.  of  the  Roy.  Stat.  Soc.  (1865),  p.  294  ff. 
Jevona  has  not  expressed  the  different  importance  of  the  different 
commodities,  i.e.,  he  has  computed  a  simple,  not  a  weighted,  geometric 
mean. 

"»  To  prove 


V 


a,  ....  an  ^  a.  +  a,  + +  an 


The  theorem  follows  by  mathematical  induction  as  follows  (changing 
the  notation)  : 

(1)   If  a  given  quantity,  a,  be   divided   into  three  parts,  x,  y,  z, 
the  maximum  value  of  the  product  xyz  is  attained  when  the 


196  THE  VARIOUS  KINDS  OF  AVERAGES 

ally  not  very  large,  especially  if  the  means  are  based  on 
a  great  number  of  items.  Thus,  the  geometric  means  of 
the  39  index  numbers  quoted  by  Jevons  for  the  years 
1851,  1853,  1855,  1857,  and  1859  amount  to  92.4,  111.3, 
117.6,  128.8,  and  116;  the  arithmetic  means  computed  from 
the  same  index  numbers  for  the  same  years  are  94.6,  112.4, 
119,  134,  and  119.«°-" 

parts  are  equal,  or  when  x  ^  y  =:  z  —  ^       This  theorem  is 

o 

easily  proven  by  the  method  of  the  differential  calculus. 

(2)  Assume  that  the  maximum  value  of  the   continued  product 

X  y  z    (to   n — 1   factors),    where  their  sum  is  a 

constant,  is  attained  when  the  factors  are  equal. 

(3)  Consider  the  product 

wx-y-z  (n  factors)  =  p 

where  w-|-x-(-y-(-z-f- rma 

If  any  value,  b,  be  assigned  to  w,  then  the  maximum  value 
of  the  product,  p,  is  attained  when  the  second  composite  factor 
is  a  maximum,  or  according  to  ( 2 ) ,  when  the  n — 1  factors  are 

a-b 

each  equal  to Consider,  therefore,  the  function  w-x°' 

n-1 
and  allow  w  to  take  values  varying  from  o  to  a.     For  what 
value  of  w  is  this  function  a  maximum? 

n-1  /a-wsn-1 

p  =  wx        =  w-  (^j-) 

for  a  maximum 

dp       /a-w\  w-2  /a-nw\  _ 

dw-v¥T/    •    \~^Ty° 

a 
w=  — 

or,  the  maximum  value  of  the  product  w-x-y-z is 

attained  when  W:=x  =  y=: =  _?_ 

n 

(4)  But,  since  the  maximum  value  of  the  product  of  three  factors 

is  attained  when  x  =  y  1=  z,  then,  by  ( 3 ) ,  the  maximum  value 
of  the  product  of  four  factors  is  attained  when  they  are  equal, 
and  so  on  indefinitely  to  n  factors. — Tkanslatob. 
*"  F.   Y.   Edgeworth,   "A  Defense  of   Index-Numbers,"  The  Econ. 
Journ.,  Vol.  VI  (1896),  p.  137. 

°'  If  we  compute  the  arithmetic,  the  harmonic,  and  the  geometric 
means  of  a  set  of  items,  then  the  last  is  at  the  same  time  the 
geometric  mean  between  the  first  two  means.     The  values  1  and  2, 


THE  GEOMETRIC  MEAN  197 

Comparisons  between  the  geometric  and  arithmetic  means 
computed  from  the  same  series  prove  that  the  former  is 
not  influenced  by  extreme  items  to  the  same  degree  as  the 
latter.  The  geometric  mean  of  commodity  index  numbers 
is,  therefore,  less  influenced  by  violent  price  fluctuations 
of  single  commodities  than  is  the  arithmetic  mean.  This 
property  of  the  geometric  mean  is  an  advantage  in  such 
problems  as  in  the  representation  of  the  movements  of  the 
price  level  where  we  do  not  think  it  justified  for  an  ex- 
ceptionally strong  change  in  the  price  of  a  single  commodity 
to  influence  the  result  to  such  a  degree  as  is  the  case  if 
the  arithmetic  mean  is  applied.  Bowley  recommends  con- 
trolling the  arithmetic  mean  by  simultaneously  computing 
the  geometric  mean  of  the  items  in  question.  If  the  arith- 
metic and  geometric  means  of  a  series  differ  considerably 
from  each  other,  the  geometric  mean  must  be  considered  to 
be  more  correct  on  account  of  the  advantage  mentioned 
above,^^ 

The  use  of  the  geometric  mean  in  computing  mean  index 
numbers  has,  as  has  been  explained  by  Westergaard,  the 
special  advantage  that  the  same  result  is  obtained  for  a 
given  period,  no  matter  if  this  period  is  taken  as  a  w^iole 
or  divided  into  shorter  epochs,  which  afterwards  are  com- 
bined. If  the  changes  of  the  price  level  from  1860  to 
1870  and  from  1870  to  1880  have  been  computed  by  use 
of  the  geometric  mean,  then  by  combining  these  changes 
the  same  result  is  obtained  for  the  price  fluctuation  from 
1860  to  1880,  as  though  the  whole  period  had  been  treated 

for  instance,  give  the  arithmetic  mean  1.50,  the  harmonic  mean  1.33, 
and  the  geometric  mean  1.41.  The  last  value  is  at  the  same  time  the 
geometric  mean  between  1.33  and  1.50.  From  this  relation  of  the 
three  means  it  follows  that  if  two  of  them  are  given,  the  third  may 
be  computed  directly  from  them  ( cf .  Messedaglia,  "  Calcul  des 
valeurs  moyennes,"  Annales  de  demographic  internationale  (18S0), 
p.  390). 

«*  Elements  of  Statistics,  2nd  ed.,  p.  128  f. 


198  THE  VARIOUS  KINDS  OF  AVERAGES 

at  once.     This  is  not  the  case  if  the  arithmetic  mean  is 

^ge(J69-60a 

"*  Cf.  Westergaard,  Die  Grundziige,  p.  218  ff. ;  see  also  Bowley, 
Elements  of  Statistics,  2nd  ed.,  p.  223. 

"a  Prof.  A.  W.  Flux  has  tested  the  effect  of  a  change  of  the  base 
year  with  reference  to  which  the  commodity  index  numbers  are 
calculated.  The  simple  or  weighted  arithmetic  mean  of  the  com- 
modity indices  was  found  to  vary  by  as  much  as  6%  on  account  of 
the  change.  Of  course  a  change  of  the  base  year  does  not  affect  the 
geometric  mean  ("Modes  of  Constructing  Index- Numbers,"  Quar. 
Joum.  Econs,,  Vol.  XXI,  p.  613.) — Tbanslatob. 


CHAPTER  IV 
THE  MEDIAN 

1.   CONCEPT  AND  PROPERTIES  OP  THE  MEDIAN 

The  median  is  that  value  which  ' '  has  the  central  position 
in  a  series  of  items  arranged  according  to  size  "  (Czuber)  ;^'* 
it  is  "  the  magnitude  appertaining  to  the  item  halfway  up 
the  series  "  (Bowley),'^^  Fechner  defines  the  median 
("  Zentralwert  ")  as  "  that  value  of  a  " — a  meaning  the 
measurements  of  any  collective  object — "  which  has  just 
as  many  values  above  it  as  there  are  below  it  and  thus 
divides  the  series  in  the  middle.  ""-73  jf  ^n  odd  number 
of  items  arranged  according  to  size  is  given,  then  the  item 
in  the  middle  of  the  series  is  the  median ;  for  instance,  of  89 
items  arranged  according  to  size  the  45th  is  the  median. 
"With  an  even  number  of  items  the  median  lies  between  the 
two  central  items.  It  has  the  same  size  as  they,  if  both  are 
equally  large ;  if  they  are  not  equal,  the  arithmetic  average 
is  usually  taken  as  the  median,  or  more  accurate  inter- 
polation may  be  used. 

The  median  differs  essentially  from  the  arithmetic  and 
geometric  means.  In  the  computation  of  the  last  two  the 
size  of  every  item  of  the  series  is  of  influence  since  these 

""  Wahrscheinlichkeitsrechung,  p.  334. 

"  Elements  of  Statistics,  2nd  ed.,  p.  124. 

'"  Kollektivmasslehre,  p.  13. 

"  The  values  which,  in  a  series  arranged  according  to  the  sizes 
of  items,  form  the  line  between  the  first  and  the  second,  and  the  third 
and  the  fourth  quarters  of  the  series  are  called  quartiles;  the  values 
dividing  the  series  into  100  parts  are  called  percentiles.  Quartiles, 
deciles,  and  percentiles  may  be  used  to  supplement  the  median. 

199 


200  THE  VARIOUS  KINDS  OF  AVERAGES 

are  added  or  multiplied  in  the  process.  The  median  is 
found  in  quite  a  different  manner.  The  median  is  a  defi- 
nite item  which,  on  account  of  its  central  position  within 
the  series,  is  considered  to  be  characteristic  of  the  series, 
or  it  is  the  average  of  the  two  items  located  in  the  middle 
of  the  series  and  therefore  considered  to  be  especially 
significant.'^*  Changes  of  the  items — except  of  the  central 
member  or  the  two  central  members — have  no  influence  at 
all  on  the  size  of  the  median,  as  long  as  the  number  of 
items  above  and  below  remains  unchanged.  If  a  series 
consists  of  the  numbers  1,  2,  3,  4,  5,  6,  7,  8,  and  9,  then 
5  is  their  median.  Any  changes  of  the  items  above  and 
below  the  median  are  entirely  without  effect,  as  long  as 
the  number  5  does  not  lose  its  central  position  in  the  series. 

On  account  of  its  independence  of  single  extreme  cases 
the  median  may  possess  a  more  "  typical  "  character  than 
the  arithmetic  mean.  If  the  arithmetic  mean  is  used  to 
represent  the  average  income  of  a  population,  then  the 
income  of  a  millionaire  will  counterbalance  the  incomes  of 
hundreds  of  workmen.  The  presence  of  a  millionaire  in  a 
district  otherwise  poor  will  give  an  arithmetic  average  in- 
come which  lies  between  the  income  of  the  mass  of  the 
population  and  that  of  the  millionaire,  a  mere  arithmetic 
abstraction,  to  which  not  a  single  real  case  cor- 
responds. If,  however,  the  median  income  is  taken  then 
the  millionaire  will  have  no  more  importance  than  any  other 
individual  and  this  average  will  undoubtedly  reflect  more 
accurately  the  income  of  the  mass  of  the  population. 

However  independent  the  median  may  be  of  extreme 
items,  it  depends  entirely  on  the  numerical  size  of  the 
central  item  or  items.  Two  series,  in  which  only  the 
central  items  differ  considerably,  while  the  other  items 
coincide    completely,    result    in    very    different    medians. 

^*  Liesse,  therefore,  correctly  calls  the  median  ("la  mMiane") 
"  une  moyenne  de  position  "  in  opposition  to  the  arithmetic  mean 
(La  Statistique,  p.  82). 


THE  MEDIAN  201 

Changes  of  the  central  item  or  items  of  a  series  may  cause 
considerable  changes  of  the  median,  even  if  all  the  other 
items  remain  unchanged.  The  comparison  of  the  medians 
alone  produces,  therefore,  a  one-sided  picture  in  cases  of 
the  kind  mentioned.  These  cases,  however,  rarely  occur 
in  practice. 

The  median  is  a  typical  value  only  if  it  appears  at  a 
point  of  concentration.  In  a  series  distributed  symmetric- 
ally around  a  point  of  concentration  it  is  identical  with 
the  arithmetic  mean  and  the  mode.  If,  however,  the  items 
are  distributed  in  such  a  way  that  a  concentration  of  the 
items  takes  place  away  from  the  center,  then,  of  course,  the 
median  has  no  typical  character.  But  the  arithmetic  mean 
of  such  a  series  would  also  be  non-typical.  In  general,  it 
may  be  said  that  most  series  can  be  characterized  by  the 
median  just  as  well  as  by  the  arithmetic  mean,  while  the 
former  has  the  advantage  that  it  is  much  easier  to  de- 
termine.^^ 

It  follows  directly  from  the  definition  of  the  median 
that  the  numbers  of  items  deviating  in  both  directions  are 
equal.  Consequently,  the  probability  that  an  item  chosen 
at  random  lies  below  the  median  is  just  as  great  as  the 
probability  that  it  lies  above.  Therefore  the  median  is 
sometimes  called  the  "  probable  "  value  of  the  element  of 
observation. ■'®  Thus  the  median  age  of  the  mortality  table 
for  all  ages  is  called  the  probable  length  of  life;  Lexis 
calls  the  median  of  the  age  constitution  of  the  living  the 
probable  age ;  '^^  Boecldi  has  called  the  duration  of  mar- 
riage expressed  by  the  median  of  his  table  of  durations 
of  marriage,  the  probable  duration  of  marriage;  in  the 
theory  of  error  the  median  of  the  series  of  errors  is  called 
the   probable   error,    etc.     When    calling   the    median   the 

'"  See  Lexis,  Zur  Theorie  der  Mnssenerscheinungen,  p.  35. 
'"  See  Czuber,  Wahrscheinliehkeitsrechnung,  p.   334,  and  Fechner, 
Kollektivmasslehre,  p.  1G6. 

^'  Zur  Theorie  der  Massenerseheinungen,  p.  36. 


203  THE  VARIOUS  KINDS  OF  AVERAGES 

probable  value,  of  course,  it  must  not  be  understood  to 
mean  that  this  is  the  most  probable  value.  The  most  prob- 
able value  is  the  one  occurring  most  frequently,  that  is,  the 
mode. 

Furthermore,  Fechner  has  proved  that  the  sum  of  the 
deviations  of  the  items  from  the  median  is  a  minimum, 
i.  e.,  smaller  than  the  sum  of  the  deviations  of  the  items 
from  any  other  value.''^ 

Accordingly  the  median  differs  mathematically  from  the 
arithmetic  mean  in  two  points.  While  the  sums  of  the 
positive  and  the  negative  deviations  from  the  arithmetic 
mean  are  equal,  the  numbers  of  the  positive  and  the  nega- 
tive deviations  from  the  median  are  equal;  while  the  sum 
of  the  squares  of  the  deviations  from  the  arithmetic  mean 
is  a  minimum,  the  sum  of  the  first  powers  of  the  deviations 
from  the  median  is  a  minimum.'^* 

2.   SERIES  IN  WHICH  THE  MEDIAN  CAN  BE  DETERMINED 

The  determination  of  the  median  is  customary  only  in 
series  of  quantitative  individual  observations  (individual 
data — for  instance,  series  of  wages  and  incomes,  ages,  etc., 
series  in  the  first  of  our  three  groups),  and  indeed,  its  use 

^*  G.  Th.  Fechner,  "  Uber  den  Ausgangswert,"  etc.,  Abhandlungen  of 
the  Saxon  Society  of  Sciences,  Vol.  XVIII,  Mathematical-physical 
group,  Vol.  XI  (1874)  ;  see  also  Lexis,  Zur  Theorie,  etc.,  p.  35,  and 
Bowley,  Elements  of  Statistics,  p.  126. 

'"  Fechner,  Kollektivmasslehre,  p.  13.  In  another  place  Fechner 
gives  the  following  example:  The  series  of  the  following  7  values 
chosen  at  random  0,  2,  4,  6,  7,  8,  and  8  gives  5  as  arithmetic  mean 
and  6  as  median.  The  sum  of  the  deviation  from  the  median  is  17, 
but  it  is  more  from  any  other  value  no  matter  whether  it  is  taken 
from  the  series  itself  or  assumed  between  any  of  the  items  of  the 
series — for  instance,  from  the  number  5  (the  arithmetic  mean),  it 
equals  18;  from  the  number  5.5,  17.5;  from  the  number  2,  23.  The 
sum  of  the  squares  of  the  deviations  is  smallest  from  the  number  5 
(the  arithmetic  mean);  in  this  case  it  is  58;  from  the  number  6 
(the  median)  it  is  65.     (See  "Uber  den  Ausgangswert,"  p.  20.) 


THE  MEDIAN  203 

is  free  from  objection  only  in  such  series,  as  will  be 
proved  in  the  following.  The  determination  of  the  median 
can  take  place  only  in  series  the  items  of  which  are 
arrayed  according  to  size.  The  members  of  the  series  of 
the  second  and  third  groups  are  usually  arranged  from 
other  points  of  view  than  according  to  size.  These  two 
groups  of  series  contain  time,  space,  qualitative,  and  quanti- 
tative series.  In  time  series  the  items  are  naturally  given 
in  chronological  order,  in  space  series  they  are  arranged 
from  some  geographical  point  of  view,  i.  e.,  according  to 
the  geographical  location  of  the  districts  in  question.  In 
qualitative  series  the  items  follow  each  other  corresponding 
to  the  logical  sequence  of  the  divisions ;  for  instance,  values 
for  different  occupations  are  arranged  according  to  the  rela- 
tions existing  between  the  groups  of  occupations  distin- 
guished. Finally,  in  quantitative  series  the  arrangement  of 
the  items  depends  on  the  gradations  of  the  quantitative  cri- 
terion used  in  the  formation  of  the  series,  consequently 
the  values  which  refer  to  different  age  classes,  for  example, 
are  given  in  the  order  of  these  age  classes.  In  order  to  deter- 
mine the  median  of  a  series  of  the  second  or  third  group  we 
would  have,  first,  to  arrange  the  items  according  to  size; 
in  order  to  accomplish  this  the  fundamental  criterion  that 
leads  to  the  formation  of  the  series  would  have  to  be  dis- 
regarded and  the  series,  therefore,  would  be  destroyed. 

There  is  another  objection  against  the  determination  of 
the  median  for  a  series  of  the  third  group,  i.  e.,  for  series  the 
members  of  which  characterize  constituents  of  a  larger 
totality  in  some  definite  manner  by  relative  numbers  or 
means.  The  members  of  such  series  (for  instance,  death 
rates  for  different  geographical  districts  or  occupations) 
usually  refer  to  constituents  of  different  sizes  and,  conse- 
quently, of  different  weights.  The  weights  of  single  mem- 
bers, however,  cannot  be  ascertained  from  the  series  itself. 
It  is  true,  we  can  arrange  the  items  of  such  a  series  accord- 
ing to  magnitude,  but  we  cannot  ascertain  the  actual  center. 


204  THE  VARIOUS  KINDS  OF  AVERAGES 

The  actual  center  of  the  series  is  located  below  or  above 
the  central  item  according  as  the  inferior  or  superior  items 
refer  to  greater  constituents  and,  consequently,  are  of 
greater  weight.^" 

From  series  of  means  we  cannot  ascertain  the  magni- 
tudes of  the  individual  observations  on  which  are  based 
the  means  forming  the  series.  Consequently  we  can  ascer- 
tain neither  the  order  of  magnitude  of  the  actual  items 
nor  their  central  member.  If  the  arithmetic  average 
wages  or  normal  wages  for  the  different  districts  of  a 
country  are  given,  it  is  impossible  to  ascertain  from  them 
the  * '  central  ' '  wage  for  the  whole  country,  i.  e.,  that  wage 
which  divides  the  individual  wages  of  the  workmen  of  the 
whole  country  in  two  equal  parts  so  that  one  half  of  the 
workmen  earn  more,  the  other  half  less.  Of  course  the 
average  or  normal  wages  may  be  arranged  according  to 
magnitude  and  the  central  member  in  this  series  of  means 
be  ascertained.  But  in  this  way  a  median  is  obtained 
which  itself  is  again  an  average.  This  median  will  not  be 
apt  to  coincide  with  that  median  which  would  result  from 
the  series  of  the  individual  wages  of  the  whole  country 
and  which  alone  could  be  considered  to  be  the  actual 
*'  central  "  wage  according  to  the  definition. 

*"  Colajanni  is  one  of  the  few  authors  who  determine  the  median 
for  series  of  relative  numbers  of  the  kind  mentioned  above.  (See 
Manuale  di  Statistica  teorica,  p.  182.)  In  the  place  cited  Colajanni 
determines  the  median  in  the  series  of  Italian  marriage  rates  for 
the  years  1872-1896  arranged  according  to  size.  We  consider  this 
procedure  to  be  theoretically  inadmissible,  since  the  marriage  rates 
mentioned  are  of  different  weights  because  the  Italian  population 
has  increased  considerably  during  the  years  in  question  and  thus 
the  data  for  the  later  years  should  have  greater  weight.  However, 
the  differences  of  weight  in  time  series  are  usually  much  smaller 
than  in  geographical  series  and  in  series  for  different  groups  of 
population,  and  therefore  the  determination  of  the  median  for  time 
series  is  less  objectionable  than  for  series  of  the  last  two  kinds 
mentioned. 


THE  MEDIAN  205 

3.   DETERMINATION  OF  THE  MEDIAN 

As  a  rule  the  determination  of  the  median  is  very  simple. 
The  central  item  of  a  series  of  an  odd  number  of  items  can 
be  found  by  merelj^  counting  the  items.  If  an  even  num- 
ber of  items  is  given,  it  is  sufl&eient  for  most  practical  pur- 
poses to  simply  take  the  average  of  the  two  central  items 
for  the  median.  If,  with  an  even  number  of  items,  we 
want  to  determine  the  median  accurately,  we  must  con- 
sider the  formation  of  the  whole  series  and  try  to  in- 
terpolate that  value  which,  according  to  the  structure  of 
the  series,  under  the  supposition  that  we  are  dealing  with 
a  continuous  variable,  would  fall  between  the  two  central 
items.  Usually,  graphical  interpolation  is  most  expedient; 
for  greater  accuracy  the  series  may  be  treated  algebraically. 

The  median  of  a  series  consisting  of  an  even  number  of 
items  suffers,  according  to  Fechner,  because  of  the  "  in- 
herent uncertainty  of  its  determination."^^  For  every 
value  between  the  two  central  items — which  Fechner  calls 
the  "  limiting  values  "  of  the  median — corresponds  to  the 
criteria  established  for  the  median.  With  every  value  be- 
tween these  limiting  values  the  number  of  deviations  on 
both  sides  is  equal  and  for  every  such  value  the  sum  of  the 
deviations  is  a  minimum,  because,  when  the  median  is  moved 
between  the  two  limiting  values,  an  increase  of  the  devia- 
tions on  one  side  is  compensated  by  a  corresponding  decrease 
on  the  other  side. 

Since  the  numerical  value  of  the  median  does  not  depend 
on  the  sizes  of  all  the  items,  but  merely  on  the  sizes  of  the 
items  located  in  the  center  of  the  series,  the  median  may 
under  certain  conditions  be  determined  for  series  in  which 
part  of  the  items  are  unknown.  The  conditions  which 
m.ust  be  fulfilled  are,  first,  that  the  number  of  the  items 
must  be  kno\\Ti  whose  individual  sizes  are  not  lvno"\^Ti,  and 
second,  it  must  be  established  that  the  sizes  of  the  items 

•'  "  Uber  den  Ausgangswert,"  p.  20  f. 


206  THE  VARIOUS  KINDS  OF  AVERAGES 

are  such  (even  though  unknown)  as  to  place  them  un- 
questionably above  or  below  the  median.  If  these  con- 
ditions are  fulfilled,  then  the  median  of  the  series  can  be 
determined  without  difficulty.  Cases  are  not  rare  where 
this  property  of  the  median  can  be  used  to  advantage.  In 
general  investigations  of  income  and  wages  certain  groups 
of  the  population  can  often  not  be  included.  The  excluded 
population  are  usually  the  classes  with  the  smallest  income 
and  the  lowest  wages  (for  instance,  persons  having  no  trade 
and  only  occasional  sources  of  income).  If  the  number  of 
persons,  about  whose  income  or  wages  no  data  can  be  ob- 
tained, is  known  approximately,  and  if  it  is  certain  that 
these  persons  belong  to  the  lowest  income — or  wage — class, 
then  the  computation  of  the  median  for  the  total  population 
is  not  impeded  by  the  fact  that  the  individual  incomes  or 
wages  of  these  lowest  classes  is  not  definitely  known.  It 
is  sufficient  that  the  number  of  persons  belonging  to  these 
classes  can  be  taken  into  consideration  when  computing  the 
median  of  the  series.^^ 

As  has  been  noted,  the  practical  statistician  frequently 
has  to  work  with  series  in  which  the  items  are  not  given 
according  to  their  exact  sizes  but  in  classes  of  a  frequency 
table.  As  has  been  explained,^^  the  limits  of  these  classes 
may  be  fixed  either  in  order  to  give  equal  frequencies  (such 
as  percentiles),  or  regardless  of  the  frequencies.  Classes 
of  the  latter  kind  usually  comprise  varying  frequencies  no 
matter  whether  they  have  equal  or  unequal  breadth. 

If  a  series  is  given  which  consists  of  an  even  number 
of  classes,  formed  according  to  the  method  of  the  percentiles, 
then  the  median  is  at  the  line  of  division  between  the  two 
central  classes  or  it  lies  in  the  middle  between  the  superior 
limit  of  the  lower  class  and  the  inferior  limit  of  the  ad- 
joining higher  class.     If  an  odd  number  of  such  classes 

*^  See,  in  this  connection,  Bowley,  Elements  of  Statistics,  2nd  ed., 
p.  125. 

«»  See  pp.  84-91. 


THE  MEDIAN  207 

is  given,  then  the  class  can  immediately  be  ascertained  in 
which  the  median  must  be  located,  but  its  numerical  size 
cannot  be  ascertained  from  the  series.  The  median  appar- 
ently lies  between  the  two  limiting  values  of  the  central 
class;  its  exact  value,  however,  can  only  be  ascertained  by 
an  exhaustive  study  of  the  series.  The  same  thing  holds 
if  a  series  consisting  of  classes  of  varying  frequencies  is 
given.  The  class  in  which  the  median  must  be  located  is 
easily  found.  Within  the  limits  of  this  class,  however,  the 
more  accurate  location  of  the  median  must  be  ascertained.^* 
The  accurate  determination  of  the  median  within  the 
limits  of  a  class  is  easily  accomplished,  if  the  original 
material  of  the  investigation  in  question  is  at  hand  and  if 
the  grouping  of  the  items  within  the  class  under  con- 
sideration can  be  examined.  This,  however,  is  not  usually 
the  case.  Therefore,  the  statistician  will  be  obliged  to  form 
a  hypothesis  as  to  the  distribution  of  the  items  within  the 
class  and  to  determine  the  median  on  the  basis  of  this 
hypothesis.  The  simplest  hypothesis  is  that  of  uniform 
distribution  of  the  items  within  the  limits  of  the  class  in 
question.  If  we  use  this  hypothesis  in  a  series  consisting 
of  an  odd  number  of  classes  of  equal  frequencies,  then,  in 
order  to  obtain  the  median,  we  have  merely  to  compute 
the  arithmetic  average  of  the  two  limiting  values  of  the 

°*  Since  the  exact  sizes  of  the  items  belonging  to  the  classes  that 
evidently  do  not  contain  the  median  have  no  influence  on  its 
numerical  size,  the  median  may  be  computed  directly  from  a  series 
in  which  the  two  end  classes  have  no  superior  or  inferior  limit, 
respectively,  while  the  computation  of  the  arithmetic  mean  of  such 
a  series  involves  considerable  difficulty.  As  has  been  mentioned 
(on  p.  145  f.),  if  a  series  of  wage  data  indicates  how  many  work- 
men belong  to  each  of  the  following  wage  classes:  less  than  $10.00, 

$10    to   $10.99,   $11    to   $11.99    $29   to   $29.99,   $30.00 

and  more,  the  median  could  be  computed  from  this  without  dif- 
ficulty, while  the  accurate  computation  of  the  arithmetic  mean  is 
bound  to  fail  because  the  sizes  of  the  items  less  than  $10  and  more 
than  $30  are  not  known. 


208  THE  VARIOUS  KINDS  OF  AVERAGES 

central  class.  If  we  have  a  series  consisting  of  classes  of 
varying  frequencies,  then,  using  the  hypothesis  mentioned, 
the  median  in  the  class  under  consideration  is  usually  de- 
termined by  dividing  the  breadth  of  the  class  in  question 
in  the  ratio  which  the  median  is  supposed  to  divide  the 
items  of  that  class.  This  method,  however,  is  not  always 
clear  or  free  from  objection,  as  the  following  example  is 
supposed  to  show. 

Let  a  series  of  wage  data  (for  instance,  weekly  wages)  for 
99  workmen  be  given.  The  wage  data  are  given  in  classes  of 
$2  breadth  each.  In  the  class  that  contains  the  wages 
from  $24  to  $25.99  there  are  10  workmen,  i.  e.,  the  48th 
to  the  57th  inclusive.  The  wage  of  the  50th  workman 
is  to  be  determined  since  this  wage  represents  the  median. 
The  50th  workman  is  the  3rd  of  the  10  workmen  belonging 
to  this  class.  Prima  facie  it  is  an  obvious  assumption 
that  his  wage  is  y^  of  the  breadth  of  the  class  higher 
than  the  wage  which  forms  the  inferior  limit  of  the  class. 
Three-tenths  of  the  breadth  gives  60  cents,  and  the  median 
would  be  $24.60.  But  the  following  argument  is  just  as 
obvious:  In  the  class  $24  to  $25.99  there  are  9  intervals 
between  the  10  items  belonging  to  this  class.  According  to 
the  hypothesis  of  uniform  distribution  these  intervals  must 
be  considered  equal.  Therefore  the  median  is  separated 
from  the  lowest  item  by  2  and  from  the  highest  by  7 
intervals.  In  order  to  determine  its  size  the  breadth  of  the 
class  must  be  divided  in  the  proportion  of  2:7.  Accord- 
ingly the  median  would  be  f  of  the  breadth  of  the  class 
above  the  inferior  limit  of  the  class  and  would  be  $24.44. 
The  inconsistency  arises  because  of  the  different  positions 
which  can  be  assigned  to  the  first  and  last  items  in  the 
class. 

We  can  distribute  10  values  in  a  class  of  200  cents 
breadth  so  that  the  first  and  the  last  values  coincide  with 
the  limiting  values  of  the  class ;  so  that  the  first  item  coin- 
cides with  the  inferior  limit  while  the  last  value  is  as  far 


THE  MEDIAN  209 

distant  from  the  superior  limit  as  are  the  items  from  each 
other;  or,  so  that  the  last  item  coincides  with  the  superior 
limit  while  the  first  item  is  as  far  distant  from  the  inferior 
limit  as  are  the  items  from  each  other.  None  of  these  three 
distributions  seems  to  be  free  from  objection.  The  first 
kind  of  distribution,  if  carried  out  in  the  adjoining  classes, 
would  give  two  items  at  each  class  limit.  The  second  and 
third  kinds  of  distribution  do  not  correspond  at  all  to  the 
postulate  of  a  uniform  distribution  within  the  classes. 
The  most  correct  way  of  distributing  the  items  uniformly 
is  to  assume  that  they  occur  at  equal  intervals  even  when 
this  distribution  is  extended  to  the  adjoining  classes.  To 
fulfil  this  condition  the  first  and  the  last  of  the  items  be- 
longing to  the  class  must  be  removed  from  the  class  limits 
to  a  distance  which  corresponds  to  half  the  magnitude  of 
the  interval  existing  between  the  items  belonging  to  the 
class.  Consequently  in  our  example  the  wages  of  the  10 
workmen  belonging  to  the  class  $24  to  $25.99  ought  to  be 
distributed  in  such  a  way  that  the  wages  increase  20  cents 
from  workman  to  workman  and  that  the  first  workman  of 
the  class  receives  $24.10  and  the  tenth  $25.90.  According 
to  this  computation  the  wage  of  the  third  workman,  and 
thus  the  median,  is  $24.50. 

The  above  principles  for  the  computation  of  the  median 
from  classes  of  a  frequency  table  must  also  be  used  if  the 
median  is  to  be  computed  from  a  series  of  measurements  for 
an  element  varying  continuously.  For  series  of  this  kind,  as 
closer  investigation  will  show,  always  consist  of  classes,  even 
though  they  seem  to  give  all  the  individual  measurements. 
Continuous  elements  of  measurement — for  instance,  dis- 
tance, area,  weight,  duration  of  time,  etc. — cannot  be  ascer- 
tained with  complete  accuracy,  and  in  statistical  measure- 
ment we  are,  as  a  rule,  satisfied  with  a  still  smaller  degree  of 
accuracy.  Individual  items,  which  coincide  in  statistical 
observation,  would  be  found  to  be  of  different  values  if 
greater  accuracy  were  used.    Therefore,  the  smallest  units 


210  THE  VARIOUS  KINDS  OF  AVERAGES 

distinguished  really  represent  classes.  How  to  proceed  in 
the  computation  of  the  median  from  data  for  a  continuous 
element  of  measurement,  may  be  explained  by  means  of  a 
series  of  measurements  of  stature,  which  are  taken  from 
Francis  Galton  in  the  Report  of  the  Anthropometric  Com- 
mittee of  the  British  Association  (1881).^^ 

The  series  in  question  gives  the  heights  of  76  boys, 
13-15  years  old.  Since  the  series  consists  of  76  members, 
the  median  lies  between  the  38th  and  39th  member.  The 
38th  as  well  as  the  39th  member  are  59  inches.  The  four 
members  preceding  the  38th  and  the  member  following  the 
39th  are  also  59  inches.  Nevertheless,  the  median  is  not 
exactly  59  inches.  Since  the  heights  have  been  recorded 
only  to  the  nearest  quarter  of  an  inch,  all  the  sizes  between 
58J  and  59^  inches  in  the  series  evidently  are  recorded 
as  59  inches.  Now  we  can  assume  that  the  7  heights  called 
59  inches  are  uniformly  distributed  between  58|^  and  59J. 
Under  this  supposition  the  38th  and  the  39th  members  lie 
between  59  and  59^;  their  accurate  position  and  the 
position  of  the  median  depend  on  the  placing  of  the  last 
member,  i.  e.,  the  40th.  If  this  be  placed  at  the  superior 
limit  of  the  class  then  the  38th  and  the  39th  members  are  -^ 
and  -ii  inch  above  59  inches.  But  if  the  items  are  dis- 
tributed so  that  the  last  of  them  is  still  a  certain  distance 
away  from  the  superior  limit,  a  distance  which  corresponds 
to  half  the  size  of  the  intervals  between  the  other  items, 
then  the  38th  and  39th  members  are  -^  and  -f^  inch  above 
59  inches.  The  median — the  average  of  these  two  members 
— is  59-^  inches.^® 

'°  These  data  given  in  Bowley's  Elements  of  Statistics  were  used 
by  Galton  to  explain  the  method  of  graphic  determination  of  the 
median,  of  which  we  shall   speak  later. 

*'  Fechner,  too,  has  examined  the  case,  in  which  the  median  "must 
be  looked  for  within  the  uniform  succession  of  the  values  of  a 
measurement  class."  (See  "  Uber  den  Ausgangswert,"  p.  18  f.)  Fech- 
ner lays  down  the  rule,  that  the  median  must  be  computed  so  that 
the  same  value  is  found  no  matter  from  which  end  we  start  to  count 


THE  MEDIAN  211 

However,  the  hypothesis  of  the  uniform  distribution  of 
the  items  between  certain  limits  is  not  always  admissible. 
On  the  contrary,  sometimes  the  structure  of  the  entire 
series  makes  it  very  probable  that  the  items  belonging  to 
a  certain  class  are  not  distributed  uniformly  within  the 
class.  If  the  class  under  consideration  is  preceded  by  a 
place  of  concentration,  starting  from  which  the  items  be- 
come less  frequent  in  both  directions,  then  it  may  be 
assumed  that  the  items  of  the  class  under  consideration 
are  more  densely  crowded  in  the  part  towards  the  place  of 
congestion  than  in  the  opposite  part.  In  estimating  the 
median  from  such  series  we  must  resort  to  hypotheses 
which  are  better  suited  to  the  formation  of  the  series  than 
the  hypothesis  of  the  uniform  distribution  of  the  items. 
Usually,  results  can  be  obtained  most  readily  by  graphic 
interpolation.  The  most  precise  determination  is  accom- 
plished by  means  of  algebraic  interpolation.  In  this,  just 
as  in  the  graphic  interpolation,  we  start  from  the  hypoth- 
esis that  the  structure  of  the  series  intimated  by  the  known 
items  holds  also  for  the  items  which  are  not  accurately 

and  interpolate.  This  self-evident  rule  has  also  been  observed  above. 
On  the  other  hand,  there  are  differences  between  the  above  discussion 
and  Fechner's  statements.  Fechner  tries  to  determine  the  size  of 
the  13th  value  in  a  class,  the  limits  of  which  are  2l^  and  Sy^  and 
to  which  16  values  belong.  To  this  purpose  he  adds  13-16  of  the 
breadth  of  the  class  to  the  lower  limit  (2^^)  and  subtracts  3-16  of 
this  breadth  from  the  superior  limit.  In  both  ways  he  finds  the 
same  value,  3.3125.  This  value  corresponds  to  the  13tli  item  only 
under  the  supposition  of  a  distribution  of  the  items  so  that  the  last 
item  coincides  with  the  superior  limit  while  the  first  item  lies  as 
much  above  the  inferior  limit  as  the  interval  between  the  other 
items  amounts  to.  Such  a  distribution,  however,  does  not  cor- 
respond to  the  idea  of  a  symmetric  distribution  of  the  items  in  the 
whole  class.  In  spite  of  this,  Fechner's  result  was  correct;  for 
although  an  even  number  of  items  was  given,  Fechner  did  not  take 
the  average  of  the  two  central  items  but  the  lower  of  these  two 
items  for  the  median.  (See  also  Fechner,  Kollekti\Tnasslehre,  p. 
168  f.) 


212  THE  VARIOUS  KINDS  OF  AVERAGES 

known,  i.  e.,  that  the  sizes  of  the  items  not  known  accu- 
rately are  such  that  they  do  not  change  the  structure  of 
the  series  based  upon  the  known  values.®^ 

Statistical  series,  both  complete  series  of  individual  ob- 
servations and  series  consisting  of  classes,  are  frequently 
adjusted  or  graduated  in  order  to  show  clearly  their  char- 
acteristic formations  and  to  remove  the  more  or  less  acci- 
dental irregularities.  In  the  adjustment  of  a  series  a 
number  of  its  items  are  more  or  less  modified.  Conse- 
quently if  the  central  items  of  the  series  are  changed  the 
adjustment  may  affect  the  numerical  size  of  the  median. 
The  median  of  the  adjusted  series  may  be  different  from 
the  median  of  the  unadjusted  series.  But  great  differences 
are  not  to  be  expected.^^ 

A  graphic  method  of  determining  the  median  (as  well 
as  the  quartiles  and  deciles)  of  a  series  has  been  explained 
by  Galton  in  the  Report  of  the  Anthropometric  Committee 
of  the  British  Association  of  the  year  1881  (p.  247).  He 
uses  the  data  above  mentioned  giving  the  stature  of  76 
boys,  13  to  15  years  old.  According  to  Bowley's  presenta- 
tion,^^ Galton 's  graphic  method  is  essentially  the  following : 
The  axis  of  abscissas  of  the  diagram  is  divided  into  equal 
parts  which  correspond  to  the  units  of  the  element  of  meas- 
urements (in  the  present  case  inches),  and  serves,  as  usual, 

*'  An  example  of  the  computation  of  the  median  of  a  series  con- 
sisting of  classes  by  means  of  algebraic  interpolation  is  found  in 
Bowley,  Elements  of  Statistics,  2nd  ed.,  p.  252  f. 

'*  It  is  a  kind  of  a  limited  adjustment  (proposed  by  Fechner  for 
a  number  of  cases  in  "  Uber  den  Ausgangswert,"  p.  23 )  if,  with  an 
odd  number  of  items  distributed  irregularly  around  the  median, 
we  take  the  arithmetic  mean  of  the  three  central  items  for  the 
median  instead  of  the  item  which  according  to  its  position  is  the 
actual  central  item;  "  generally  we  may  feel  assured  of  coming  closer 
to  the  true  median,  which  would  be  obtained  from  an  infinite  number 
of  items,  by  using  this  method,  than  by  using  the  item  which  is 
influenced  by  accidental  errors." 

«» Elements  of  Statistics,  2nd  ed.,  p.  127  f. 


THE  MEDIAN  213 

to  represent  the  different  measurements  occurring  in  the 
series.  To  express  the  frequency  of  these  different  magni- 
tudes ordinates  are  used,  but  in  a  way  differing  from  the 
usual.  For  it  is  characteristic  of  Galton's  method  that 
the  ordinates  belonging  to  the  different  magnitudes  are  not 
all  placed,  as  usual,  vertically  on  a  common  basis  (the 
axis  of  abscissas),  but,  when  placing  successive  ordinates, 
he  measures  from  a  new  base  at  the  height  of  the  upper 
end  of  the  preceding  ordinate.  Thus,  it  results  that  the 
upper  end  of  the  ordinate  of  the  last  (highest)  measure- 
ment occurring  is  just  as  many  units  on  the  axis  of  ordi- 
nates above  the  axis  of  abscissas  of  the  diagram,  as  the 
series  has  items.  Each  ordinate  thus  represents  the  num- 
ber of  items  with  a  measurement  equal  to  or  less  than  the 
corresponding  abscissa.  Then  Galton  connects  the  upper 
ends  of  the  ordinates  and  a  broken  line  originates  on  which 
a  definite  point  corresponds  to  every  magnitude  occurring 
on  the  axis  of  abscissas.  Finally,  he  bisects  the  last  ordinate 
and  draws  a  horizontal  line  through  such  point  parallel  to 
the  base.  The  abscissa  of  the  point  of  intersection  of  the 
horizontal  and  the  broken  line  is  the  median.  In  a  similar 
way  the  quartiles  and  deciles  of  the  series  may  be  found. 

By  joining  the  tops  of  the  ordinates  with  straight  lines 
it  is  assumed  that  the  distribution  of  the  items  within  the 
classes  is  uniform,  and  the  values  of  the  quartiles  and 
deciles  found  by  such  graphic  interpolation  depend  upon 
that  assumption.  We  may  also  connect  the  tops  of  ordi- 
nates by  as  regular  a  curve  as  possible,  or  draw  a  curve 
which  (even  if  it  does  not  contain  all  the  points,  neverthe- 
less passes  very  close  to  them)  fits  the  configuration  of  the 
broken  line  as  closely  as  possible  and  clearly  represents 
the  formation  of  the  series  intimated  in  the  diagram.  From 
such  an  adjusted  curve  the  median  may  be  determined  as 
explained  in  the  preceding  paragraph. 

Galton's  method  of  the  graphic  determination  of  the 
median  is  the  counterpart  of  his  *'  cumulative  "  groups. 


214  THE  VARIOUS  KINDS  OF  AVERAGES 

These  groups  are  obtained  by  successively  summing  the  fre- 
quencies as  given  in  the  usual  table,  starting  with  the  lowest 
or  highest  class.®"  In  the  graphic  method  of  Galton,  it  is 
true,  no  numerical  sums  were  ascertained  arithmetically,  but 
as  every  ordinate  commences  at  the  elevation  of  the  end  of 
the  preceding  ordinate,  the  total  ordinates  (measured  from 
the  basis  of  the  diagram)  show  how  many  items  belong 
to  every  given  class  together  with  those  belonging  to  all 
the  preceding  classes.  It  is  a  graphic  representation  of 
the  numerical  sums  which  would  be  found  if,  starting  from 
the  lowest  class,  the  single  classes  were  added  successively. 

Bowley  has  applied  Galton 's  method  to  the  determination 
of  the  median  from  a  series  of  wages.®^  In  the  diagram 
constructed  by  Bowley,  the  abscissas  correspond  to  the 
different  wages  under  consideration,  the  ordinates  do  not 
correspond  to  the  numbers  of  workmen  with  wages  of  a 
certain  size,  but  to  the  numbers  of  workmen  earning  at  or 
above  the  wage  represented  by  the  abscissa.  The  line  join- 
ing the  tops  of  the  ordinates  has  its  lowest  point  in  the 
highest  wage  class  at  the  right  of  the  diagram,  where  there 
are  only  a  few  workmen  who  earn  that  wage  or  more.  The 
line  ascends  continuously  toward  the  left  end,  for  the  lower 
the  wages  the  greater  is  the  number  of  workmen  earning 
at  or  above  this  wage.  Since  the  height  of  the  diagram 
corresponds  to  the  total  number  of  items,  therefore,  by 
bisecting  this  height,  we  can  immediately  ascertain  that 
point  of  the  diagram  the  abscissa  of  which  represents  the 
median. 

Bowley  has  also  adjusted  the  diagram  just  described. 
The  median  determined  from  the  unadjusted  diagram  was 
$1.49,  the  same  as  the  median  computed  by  the  elementary 
arithmetic  method  from  the   original  series  of  numbers. 

*"  See  details  on  this  method  of  formation  of  groups,  p.  85  f. 

«'  Elements  of  Statistics,  2nd  ed.,  p.  154  f.  The  series  which  he 
uses  represents  the  wages  of  5,123  American  workmen  taken  from 
U.  S.  Senate  Report  on  Wages,  Prices,  etc.    (1893). 


THE  MEDIAN  215 

From  the  adjusted  curve  the  median  $1.51  was  obtained; 
mathematical  interpolation  gave  $1,536. 


4.   APPLICATION  OF   THE   MEDIAN 

The  application  of  the  median  to  represent  statistical 
series  is  far  less  extensive  than  that  of  the  arithmetic  mean. 
However,  it  is  used  in  various  fields  of  statistics  to  represent 
important  quantitative  phenomena. 

In  the  field  of  population  statistics  the  median  frequently 
serves  to  represent  the  age  constitution  of  various  masses 
of  population.  The  most  important  application  of  the 
median  in  this  field  is  the  "  probable  lifetime  "  (wahr- 
scheinliche  Lebensdauer,  vie  probable  or  vie  mediane,  vita 
probabile  or  vita  mediana),  which  is  the  median  computed 
from  the  length  of  lives  given  in  the  mortality  table  for 
a  definite  generation.  The  probable  lifetime  tells  what 
age  half  the  individuals  of  a  generation,  born  contempo- 
raneously, will  survive.  It  may  be  ascertained  for  the 
new-born  as  well  as  for  persons  in  other  age  classes.  Since 
the  mortality  tables  usually  contain  age  classes  of  one  year 
each,  the  principles  indicated  above  for  the  determination 
of  the  median  in  a  series  consisting  of  classes  must  be  ob- 
served in  the  computation  of  the  probable  lifetime. 

The  probable  lifetime  of  a  population  is  usually  longer 
than  the  expectation  of  life.  The  reason  for  this  is  that 
the  very  high  death  rates  in  the  age  classes  above  the 
probable  lifetime  remain  without  influence  upon  the  prob- 
able lifetime  (the  median),  while  they  must  essentially 
reduce  the  expectation  of  life  (the  arithmetic  mean).®^ 
The  probable  lifetime  does  not  express  at  air  the  mortality 
in  the  older  age  classes,  since,  according  to  the  nature  of 
the  median,  the  numerical  values  of  the  items  located 
away  from  the  center  of  the  series  have  no  influence  upon 
the  magnitude  of  the  median.  Therefore,  the  probable  life- 
»'  Cf.  V.  Mayr,  Bevolkerungsstatistik,  p.  268. 


216  THE  VARIOUS  KINDS  OF  AVERAGES 

time  may  remain  the  same  even  if  the  mortality  of  the 
older  age  classes  changes  considerably.  The  expectation  of 
life,  on  the  contrary,  being  the  arithmetic  mean  of  the  life- 
times under  consideration,  depends  on  the  values  of  all 
the  items  contained  in  the  series;  therefore,  it  is  influenced 
by  the  mortality  in  the  highest  classes  and  reflects  every 
change  in  them  as  well  as  changes  in  other  classes. 

Of  special  influence  on  the  probable  lifetime  is  the  in- 
tensity of  the  mortality  of  children.  If  the  mortality  of 
children  in  the  first  year — as  it  happens  in  some  districts — 
is  50^  of  those  born,  then  the  probable  lifetime  of  the 
newborn  is  one  year.^^  The  expectation  of  life  is,  of 
course,  much  higher,  since  it  depends  also  on  the  numerical 
values  of  the  lifetimes  of  those  persons  not  having  died  in 
childhood.  In  contrast  to  the  intensity  of  the  mortality 
of  children,  the  distribution  of  this  mortality  over  the 
different  age  classes  has  no  influence  upon  the  probable  life- 
time, of  course,  on  the  condition  that  the  probable  lifetime 
itself  is  not  located  in  the  younger  age  classes.  Therefore, 
the  probable  lifetime  expresses  neither  changes  in  the  mor- 
tality of  the  young  age  classes  nor  of  the  older  age  classes. 

Sometimes  the  median  is  also  computed  from  the  ages 
given  in  death  registers.  In  former  times  this  median  fre- 
quently was  called  the  probable  lifetime,  just  as  the  arith- 
metic mean  of  the  ages  of  the  dead  was  called  the  expecta- 
tion of  life.  However,  it  is  now  an  established  fact  that 
the  mean  and  the  probable  duration  of  life  can  be  computed 
correctly  only  from  mortality  tables. 

It  is  self-evident  that  the  median  may  also  be  computed 
from  the  age  constitution  of  those  living.  Lexis  calls  this 
median  the  "  central  age  "  and  thinks  that  this  value  is 
just  as  well  adapted  for  the  general  characterization  of 
the  age  conditions  of  the  population  of  a  country  as  the 
arithmetic  mean  age  of  those  living,  which  can  be  ascer- 
tained only  after  tedious  arithmetic  operations.     Further- 

»'Ibid.  p.  267. 


THE  MEDIAN  217 

more,  the  chances  are  even  that  the  age  of  an  individual 
chosen  at  random  from  the  population  exceeds  (or  is  less 
than)  the  central  age.  Therefore,  the  central  age  may  be 
called  the  probable  age  of  those  living  contemporaneously.** 

Likewise,  the  median  of  the  ages  of  those  marrying  is  of 
interest.  It  may  be  called  the  probable  marrying  age. 
Boeckh  has  also  determined  the  probable  duration  of  mar- 
riage (the  median)  from  his  frequency  table  of  marriages 
in  Berlin  according  to  their  duration. 

The  application  of  the  median  has  recently  been  extended 
to  the  field  of  wage  statistics.  Fox  was  the  first  to 
use  the  median  of  wages  extensively  in  his  Beport  on 
the  Wages  and  Earnings  of  Agricultural  Laborers  in  the 
United  Kingdom  (1900).  In  that  report  (p.  25)  the 
median  wage  rate  is  defined  as  the  *'  rate  so  chosen,  that 
the  numbers  of  laborers,  whose  rates  of  wages  are  above 
and  below  that  rate,  respectively,  are,  as  nearly  as  possible, 
equal."  In  the  Journal  of  the  Boyal  Statistical  Society  ^^ 
an  anonymous  reviewer  of  the  report  (Bowley?)  welcomed 
the  use  of  the  median  with  the  following  words:  "  All 
Statisticians  will  be  glad  to  see  this  most  useful  average 
at  last  boldly  and  explicitly  used  in  an  official  publication." 
Also  in  the  Second  Report  by  Mr.  Wilson  Fox  on  the 
Wages,  Earnings,  and  Conditions  of  Employment  of  Agri- 
cultural Laborers  in  the  United  Kingdom  (1905)  median 
wage  rates  are  presented  in  a  similar  way  as  in  the  first 
report.**' 

•*  Lexis,  Zur  Theorie  der  Massenerscheinungen,  p.  36. 

»=  Vol.  LXIII   (1900),  p.  505. 

•°  In  the  first  report  the  median  rate  was  also  called  the  pre- 
dominant rate.  To  this  name  the  reviewer  in  the  Journal  of  the 
Roy.  Stat.  Soc.  had  justly  objected,  since  by  this  term  we  denote 
the  most  frequent  wage  ( the  mode  in  the  series  of  wage  data ) .  But, 
obviously,  the  relatively  most  frequent  and  the  median  wage  need 
not  coincide.  In  the  second  report  the  term  predominant  rate  for 
the  median  wage  is  omitted.  However,  it  is  stated  that  the  median 
rate  "  in  almost  every  country  corresponded  to  the  predominant  rate. 


218  THE  VARIOUS  KINDS  OF  AVERAGES 

The  median  is  also  used  extensively  in  the  wage  statistics 
of  the  United  States  Bureau  of  the  Census  of  the  year 
1903,^^  and  that  principally  (in  Chapter  II)  to  compare  the 
wage  conditions  in  the  years  1890  and  1900.  The  quartiles 
of  the  series  in  question  are  used  here  as  complements  of 
the  medians.  "When  computing  the  median  the  problem 
was  to  find  the  ' '  employee  who  stands  halfway  between  the 
lowest-paid  and  the  highest-paid  employee. ' '  ^^  The  wage- 
class  to  which  this  employee,  standing  in  the  center  of  the 
series,  belongs,  is  easily  found.  However,  in  the  statistics 
quoted  the  particular  wage  of  the  halfway  individual  is 
not  ascertained  within  the  limits  of  the  wage  class  under 
consideration,  but  the  inferior  limit  of  that  wage  class  is 
taken  as  median.  Although  this  is  a  simplified  procedure 
it  is  not  theoretically  correct.®^ 

It  is  possible  that  the  median  will  be  more  widely  used 
in  the  field  of  wage  statistics  in  the  future.  Bowley  favors 
the  use  of  the  median  and  illustrates  the  methods  of  the 
determination  of  the  median  usually  by  series  of  wage 
statistics."" 

that  is  the  rate  at  which  the  largest  number  of  laborers  in  each 
county  was  paid"  (p.  27  and  p.  148).  It  is  not  clear  in  what  way 
the  median  wage  rates  in  the  two  reports  were  computed  in  those 
cases  where  the  median  served  to  characterize  the  wages  of  a  whole 
county,  the  computation  being  based  upon  wage  data  for  various 
rural  districts.  In  these  cases  individual  wage  data  were  not  given, 
but  merely  "  rates  of  weekly  cash  wages  most  generally  paid  to 
ordinary  agricultural  laborers "  for  the  single  rural  districts,  into 
which  counties  are  divided.  Thus,  series  of  averages  (modes)  were 
given,  from  which  correct  medians  cannot  be  computed  for  reasons 
which  have  been  explained  in  another  place   (p.  204  f.). 

"'  Twelfth  census  of  the  United  States,  special  report,  Employees 
and  Wages. 

•'  Ibid.  p.  xxvi. 

'"  Ibid.  p.  xxxi. 

100  Prof.  Mandello,  as  he  announced  in  the  Bulletin  de  I'lnstitut 
intern,  de  Statistique,  Vol.  XIII,  No.  1,  p.  401,  has  used  the 
median  in  the  field  of  historical  wage  statistics,  published  in  the 
Hungarian  language. 


THE  MEDIAN  219 

Of  the  other  fields  of  statistics  where  the  median  is  used, 
that  of  anthropological  statistics  is  the  most  important. 
If  it  is  desired  to  quickly  characterize  a  series  of  anthropo- 
metric data  by  a  mean,  the  median  is  most  available.  Since 
anthropometric  series  frequently  have  a  regular,  symmetric 
structure  in  which  the  arithmetic  mean,  the  median,  and 
the  mode  lie  close  together,  every  one  of  these  values  has  a 
typical  character.  The  median,  however,  has  the  special 
advantage  that  it  can  be  determined  the  easiest  and  the 
quickest,  i.  e.,  by  mere  counting. 

Nor  in  the  field  of  price  statistics  is  the  median  altogether 
unknown.  It  has  been  used  in  the  computation  of  mean 
index  numbers,  i.  e.,  in  the  computation  of  a  mean  of  the 
special  indices  representing  the  price  fluctuations  of  differ- 
ent commodities  in  the  course  of  years.^''^"^*'^ 

Finally,  we  may  refer  to  the  importance  of  the  median 
in  the  theory  of  error,  both  in  its  application  to  actual 
errors  of  observation  and  to  statistical  data.    In  the  theory 

""  Bowley  recommends  the  use  of  the  median  in  the  computation 
of  total  index  numbers,  especially  on  account  of  its  independence 
of  extreme  values  which  may  originate  from  abnormal  price  fluctua- 
tions of  single  commodities   (Elements  of  Statistics,  2nd  ed.,  p.  224). 

^"^  See  also  W.  C.  Mitchell,  Gold,  Prices  and  Wages  under  the 
Greenback  Standard,  for  further  use  of  the  median, 

Irving  Fisher  has  adopted  the  median  in  his  study,  The 
Purchasing  Power  of  Money.  He  agrees  with  Edgeworth's  con- 
clusion that  "  in  the  present  state  of  our  knowledge,  and  for  the 
purposes  on  hand,  the  median  is  the  proper  formula"  (Edgeworth, 
"  First  Report  on  Monetary  Standard,"  Report  of  the  British  Asso- 
ciation for  the  Advancement  of  Science,  1887,  p.  191).  After  making 
various  tests  of  the  reliability  of  the  median  Prof.  Fisher  says, 
"  The  final  practical  conclusion,  therefore,  is  that  the  weighted 
median  serves  the  purposes  of  a  practical  barometer  of  prices, 
and  also  of  quantities,  as  well  as,  if  not  better  than,  formulae 
theoretically  superior"  (Tlie  Purchasing  Power  of  Money,  p.  427). 
Edgeworth  gives  many  arguments  for  the  use  of  the  median  in  the 
reference  cited  above.  HoAvever,  Francis  Galton  was  the  real  popu- 
larizer  of  the  median  (see  Natural  Inheritance,  p.  47). — Teanslator. 


220  THE  VARIOUS  KINDS  OF  AVERAGES 

of  error  the  median  is  determined  for  the  series  of  the 
deviations  of  the  items  from  the  mean  and  with  other 
averages  of  these  deviations  serves  as  a  measure  of  the 
dispersion  of  the  series.  The  median  of  the  deviations  of 
the  items  from  the  mean  is  called  "  probable  deviation  " 
(or  "  probable  error  ")  and  it  is  that  deviation  which  is  ex- 
ceeded just  as  often  as  it  is  not  exceeded,"-''  so  that  there 
is  the  same  probability  that  an  item  chosen  at  random  has 
a  smaller  deviation,  as  there  is  that  it  has  a  greater  devia- 
tion than  has  the  median.^"^ 

A  peculiarity  of  the  median,  to  which  Galton  first  called 
attention,^"*  is  found  in  the  fact  that  it  can  also  be  com- 
puted if  different  size — or  intensity — scales  of  a  non- 
measurable  phenomenon  are  given.  Of  such  series  no  other 
average  but  the  median  can  be  computed.  An  illustration 
follows. 

Let  the  problem  be  to  ascertain  the  average  intelligence 
of  a  group  of  students.  The  different  degrees  of  intelli- 
gence cannot  be  given  numerically.  But  often  it  is  not 
difficult  to  arrange  the  students  according  to  the  degree 
of  their  intelligence.  Then  the  student  standing  in  the 
center  of  the  series  represents  the  median  intelligence  of 
that  group  of  students.^"^ 

In  this  way  it  would  also  be  possible  to  compare  the 
mental  qualities  of  various  groups  of  students  who  differ 
from  each  other  in  sex,  social  standing,  descent,  etc.  Some- 
thing definite  might  be  established  by  this  method,  espe- 

"''a  Lexis,  Zur  Theorie,  etc.,  p.  24. 

*•■  Since  one-quarter  of  the  observations  lie  on  each  side  of  the 
mean,  i.  e.,  within  the  limits  of  the  probable  error,  Yule  and  Bowley 
also  call  the  latter  the  quartile  deviation. 

^"^  Cf.  Natural  Inheritance,  p.  47,  and  "  Statistics  by  Inter-com- 
parison," Philosophical  Magazine,  Vol.  XXXIX  (January,  1875), 
p.  33. 

*•"  Cf.  Bowley,  Elements  of  Statistics,  2nd  ed.,  p.  126;  Lexis,  Zur 
Theorie,  etc.,  p.  39  f. ;  Lexis,  Abhandlungen,  etc.,  VI,  "  The  Typical 
Values  and  the  Law  of  Error,"  p.  126. 


THE  MEDIAN  221 

cially  about  the  relative  intelligence  of  the  two  sexes. ^"" 
The  median  can  also  be  computed  in  series  which  do  not 
give  the  size  or  intensity  of  a  phenomenon  numerically  but 
by  descriptive  terms.  Bowley  gives  an  example  of  the  ap- 
plication of  the  median  (with  simultaneous  utilization  of 
the  quartiles)  in  the  representation  of  such  a  series  of  non- 
numerical  data.^"^ 

The  point  in  question  is  to  characterize  by  an  average  the 
information  about  the  extent  of  working  overtime,  the 
information  being  given  by  88  branches  of  the  Amalgamated 
Society  of  Engineers  of  20,666  members.  The  data  fur- 
nished by  the  branches  only  rarely  give  the  numerical 
amount  of  overtime.  Most  of  the  answers  are  without 
numerical  precision,  such  as  "  very  little,"  "  when  nec- 
essary," "  moderate,"  '*  rather  general,"  etc.  Bowley 
arranges  these  answers  according  to  the  extent  of  overtime 
indicated  by  them  and  selects  the  answer  located  in  the 
center  of  the  series.  The  extent  of  overtime  given  by  this 
answer  may,  in  a  certain  sense,  be  considered  to  be  the 
average.  When  determining  this  central  item,  account  had 
to  be  taken  of  the  fact  that  the  individual  answers  refer 
to  varying  numbers  of  workmen,  since  the  individual  branch 
societies  have  varying  membership.  The  series  corresponds 
to  a  series  of  wage  data  in  which  varying  numbers  of  work- 
men belong  to  the  different  wage  classes  established.  Tak- 
ing into  consideration  the  varying  membership  Bowley 
found  the  statement  ''  maximum  18  hours  in  4  weeks  "  or 
"  moderately  "  to  be  the  median  for  the  characterization 
of  the  extent  of  overtime  work;  the  lower  quartile  was 
' '  very  little, ' '  the  upper  '  *  14  hours  when  busy. ' '  "Without 
consideration  of  the  varying  membership  the  median  ' '  not 
much,"  and  the  quartiles  "  very  little  "  and  '*  when  nec- 
essary "  or  "  occasionally  "  were  found. 

^"^  Cf.  Lexis,  "  Anthropologie  und  Anthropometric"  in  the  Handw. 
d.  Staatsw.,  2nd  ed.,  p.  393  f. 
^'>''  Elements  of  Statistics,  2nd  ed.,  p.  136  S. 


CHAPTER  V 
THE  MODE 

1.   CONCEPT  AND  PROPERTIES  OF  THE  MODE 

The  mode,  the  predominant,  most  usual,  or  normal  value, 
the  mean  of  density,  or  the  place  of  greatest  density 
(German:  "  der  dichteste  Wert,"  French:  usually 
"  valeur  normale  "  or  simply  "normale, "  also  "modus"), 
is  the  value  occurring  most  frequently  in  a  series  of  items 
and  around  which  the  other  items  are  distributed  most 
densely.  Fechner  defines  the  mode  as  that  value  "  around 
which  the  items  and,  consequently,  the  deviations  collect 
most  densely,  so  that  equal  intervals  contain  more  items 
the  nearer  the  intervals  lie  to  this  value,  no  matter  if 
they  are  taken  on  the  positive  or  the  negative  side."^"^ 
Therefore,  the  mode  represents  the  most  probable  value  of 
the  element  of  observation  represented  in  the  series.  * '  The 
mode  lies  at  a  place  of  concentration  in  the  series  arranged 
according  to  the  sizes  of  the  items,  so  that  the  probability 
that  an  item  chosen  at  random  will  belong  to  a  group  of 
values  which  contains  the  mode  is  greater  than  that  it  will 
belong  to  any  other,  equally  large,  group  of  items  " 
(Czuber)."» 

If  a  series  of  individual  observations  is  represented  graph- 
ically, as  usual,  by  a  system  of  coordinates  so  that  the 
abscissas  of  the  single  points  of  the  diagram  correspond 
to  the  various  magnitudes  occurring  in  the  series,  and  the 

^°^"Uber  den  Ausgangswert,"  p.   11. 

^"^  Wahrscheinlichkeitsrechnung,  p.  334.  See  also  Lexis,  Zur 
Theorie,  etc.,  p.  27;  also  Fechner,  Kollektivmasslehre,  p.  171. 

222 


THE  MODE  223 

ordinates  represent  the  frequencies  of  the  dijfferent  mag- 
nitudes, then  the  mode  of  the  series  is  the  abscissa  of  the 
maximum  ordinate  of  the  diagram.  On  account  of  this 
property  the  mode  is  sometimes  called  the  '  *  maximum  ordi- 
nate average."  ^"®* 

The  mode  has  many  properties  in  common  with  the 
median.  It  is,  just  as  the  latter,  "  une  moyenne  de  posi- 
tion," i.e.,  its  size,  unlike  that  of  the  arithmetic  and 
geometric  means,  does  not  depend  on  the  sizes  of  all  the 
items,  but,  like  the  median,  merely  on  the  sizes  at  a  definite 
place  of  the  series.  Just  as  the  median  is  given  by  the 
item  in  the  center  of  the  series,  so  the  mode  is  given  by 
the  size  of  that  item  which,  on  account  of  its  relatively 
greatest  frequency,  is  considered  characteristic  of  the  whole 
series.  The  numerical  sizes  of  the  items  that  are  distant 
from  the  place  of  concentration  containing  the  mode  are 
disregarded  and  the  mode  gives  no  information  about  the 
sizes  of  these  items.  Therefore,  series  in  which  the  places 
of  concentration  coincide  give  the  same  modes  even  if 
the  parts  of  the  series  above  and  below  the  places  of  con- 
centration are  entirely  different  in  the  two  series.  Changes 
that  occur  in  the  course  of  time  in  the  parts  of  series  aside 
from  the  place  of  concentration  do  not  influence  the  size 
of  the  mode. 

Just  as  the  median,  so,  for  similar  reasons,  the  mode 
can  only  be  computed  for  series  of  quantitative  individual 
observations,  such  as  series  of  wages,  incomes,  ages,  etc. 
(series  of  the  first  of  the  three  groups).  It  is  only 
in  series  of  this  kind  that  the  items  are  arranged  according 
to  value,  which  arrangement  is  required  in  determining 
the  mode.  In  the  series  of  the  third  group,  consisting  of 
relative  numbers  or  means,  the  mode  cannot  be  computed 
because  the  items  of  these  series  usually  refer  to  varying 
numbers  of  units  and,  therefore,  are  of  different  weights, 

losaVenn  used  this  term  (see  article  on  "Average"  in  Palgrave's 
Diet.  Pol.  Econ.),  but  his  usage  has  not  been  adopted. — Tbanslatob. 


224  THE  VARIOUS  KINDS  OF  AVERAGES 

these  weights,  however,  not  being  ascertainable  from  the 
series  itself.  Therefore,  the  actual  distribution  of  the  vari- 
ous intensities  and  sizes  cannot  be  found  from  the  series, 
and  tlie  intensity  or  the  size  actually  occurring  most  fre- 
quently cannot  be  ascertained.  If,  for  example,  a  series 
of  death  rates  referring  to  different  geographical  districts 
is  given,  then  the  rate  most  frequently  occurring  in  the 
series  may  actually  pertain  to  a  smaller  number  of  people 
than  some  other  rate  occurring  less  frequently  in  the  series. 
That  will  be  the  case  if  the  former  rates  are  based  on 
smaller,  and  the  latter  rates  on  larger,  geographical  dis- 
tricts."" 

As  far  as  series  of  averages  are  concerned,  the  sizes  of 
the  individual  observations,  on  which  the  averages  forming 
the  series  are  based,  cannot  be  ascertained  from  them.    But 

^^*  Colajanni  is  one  of  the  few  statisticians  who  try  to  determine 
the  mode  also  in  series  of  relative  numbers  of  the  kind  mentioned 
above  (see  Manuale  di  Statistiea  teorica,  Napoli,  1904,  p.  182  ff.). 
He  distinguishes  between  the  "  ordinata  massima  "  and  the  "  media 
di  densita."  By  the  first  term  Colajanni  means  the  real  mode  of 
a  series  of  individual  observations  (for  instance,  measurements, 
wages ) .  By  "  media  di  densita,"  however,  he  means  some  sort  of 
a  mode  of  a  series  of  relative  numbers.  Colajanni  gives  an  example 
of  the  computation  of  a  "  media  di  densita  "  from  classes  of  equal 
breadth  in  a  series  of  marriage  rates  for  the  years  1872-1879,  arranged 
according  to  size.  He  obtained  his  result  by  computing  the  arithmetic 
mean  of  the  class  of  greatest  frequency.  The  author,  however, 
raises  the  objection  that  series  of  relative  numbers,  for  instance, 
marriage  rates,  do  not  admit  the  computation  of  a  correct  mode  for 
the  reasons  just  mentioned,  but  especially  on  account  of  the  different 
weights  of  the  items.  The  Italian  marriage  rates  which  Colajanni 
uses  are  of  different  weights,  because  the  Italian  population  has 
changed  numerically  in  the  course  of  years.  The  marriage  rates  of 
the  later  years  have  greater  weight  on  account  of  the  growth  of 
population.  The  degree  of  intensity  occurring  most  frequently  in  the 
series,  if  the  values  in  question  refer  to  the  earlier  years,  may  be 
of  less  importance,  for  the  purpose  of  computing  an  average,  than 
a  degree  of  intensity  occurring  less  frequently  in  the  series,  the 
values  of  which  are  based  on  the  larger  population  of  the  later  period. 


THE  MODE  225 

the  sizes  of  these  individual  observations  are  necessary  if 
we  want  to  determine  the  most  frequent  individual  item. 
If,  for  instance,  a  series  of  average — or  normal — wages  for 
the  different  districts  of  a  country  is  given,  then  it  is  im- 
possible to  determine  from  these  values  the  relatively  most 
frequent,  densest,  normal  wage  for  the  whole  country,  i.  e., 
that  wage  which  relatively  most  individuals  earn.  It  is 
true  we  can  arrange  the  average  or  normal  wages  referring 
to  the  various  districts  according  to  their  sizes  and  see  if 
this  series  contains  a  place  of  concentration.  But  the  mode 
eventually  found  in  this  way  does  not  need  to  coincide 
at  all  with  that  mode  which  could  be  determined  from  the 
unknown  series  of  individual  wages  of  the  workmen  of 
the  whole  country  and  which  alone  could  be  considered 
to  be  the  actual ' '  densest  ' '  wage  for  the  whole  country. 

Finally,  we  must  recognize  that,  owing  to  its  nature,  the 
mode  is  significant  only  in  series  that  consist  of  a  large 
number  of  items  and,  therefore,  can  contain  a  "  place 
of  concentration  "  in  the  true  sense  of  the  phrase.  This 
condition  is  true  only  in  series  of  individual  observations 
(series  of  the  first  group)  ;  the  series  of  the  second 
and  third  groups  usually  consist  of  but  relatively  few 
items. 

The  mode  is  not  a  power  mean  in  Fechner's  sense.  It 
was  pointed  out  in  defining  the  mode  that  this  mean 
represents  the  most  probable  value.  The  fact  that  this 
mean,  in  contrast  to  the  arithmetic  mean  and  to  the 
median,  can  never  be  a  mere  arithmetic  abstraction  and 
always  possesses  more  or  less  typical  value,  is,  however,  of 
greater  practical  importance.  Owing  to  its  conception,  the 
mode  always  represents  a  quantity  which  occurs  with  rela- 
tively the  greatest  frequency  in  the  series  of  items  and 
around  which  the  other  items  are  grouped  ^A-ith  more  or 
less  regularity.  It  is,  of  course,  useful  to  know  what  wage 
relatively  most  workmen  earn,  at  what  age  relatively  most 
people  die,  marry,  etc.     Changes  of  the  mode  in  course 


226  THE  VARIOUS  KINDS  OF  AVERAGES 

of  time,  in  contrast  to  changes  of  the  arithmetic  mean  or 
the  median,  always  mean  changes  in  which  a  greater  num- 
ber of  units  of  observation  take  part.  If  the  arithmetic 
mean  of  a  series  of  wage  data  is  computed,  then  the  dis- 
appearance of  a  few  extreme,  high  or  low,  wage  items  may 
cause  considerable  change  in  the  average.  But  a  change  of 
the  modal  wage,  i.  e.,  of  that  wage  which  relatively  most 
wage-earners  receive,  cannot  occur,  unless  the  wages  of 
numerous  individuals  have  changed.  However,  if  the  mode 
has  increased  or  decreased  a  certain  amount,  it  does  not 
follow  that  the  same  individuals  are  necessarily  in  both 
modal  groups,  and  that  they  have  all  experienced  personally 
the  observed  change  of  the  densest  wage. 

But  the  mode  is  not  merely  of  interest  with  reference 
to  the  items  which  it  characterizes  directly.  The  impor- 
tance of  the  mode  lies  in  the  fact  that  it  is  the  average  best 
suited  to  represent  the  ''  normal  "  or  "  typical  "  size  of  a 
variable  phenomenon.  In  the  sense  of  mathematical  sta- 
tistics, it  is  true,  we  can  speak  of  a  typical  average  of 
individual  observations  only  if  the  items  show  a  grouping 
around  the  average  which  corresponds  to  the  hypothesis 
of  merely  accidental  disturbances  of  a  normal  value.^^^ 
However,  such  typical  means,  strictly  speaking,  occur  very 
seldom.  The  practical  statistician  must  take  into  account 
the  fact  that  most  statistical  series  do  not  show  a  regular 
structure  corresponding  to  the  law  of  chance  and,  conse- 
quently, have  no  typical  means  in  the  strict  mathematical 
sense. 

Of  the  averages  computed  from  statistical  series,  the 
arithmetic  mean  and  the  median  very  often  are  not  typical. 
Frequently,  they  are  values  which  do  not  at  all,  or  only  very 
rarely,  appear  in  the  series.  However,  the  mode  lies  at  a 
place  of  concentration  around  which  the  series  is  distributed 
in  both  directions  with  some  regularity.  It  may  be  as- 
sumed that  it  comes  nearest  to  that  value  which  the  general 

*"  See  Lexis,  Zur  Theorie  der  Massenerscheinungen,  p.  38. 


THE  MODE  227 

complex  of  causes  would  produce  if  free  from  disturbing 
influences.  In  non-mathematical  statistical  terminology  the 
mode  is,  therefore,  frequently  called  "  typical  "  or  ''  nor- 
mal "  value  even  when  the  distribution  does  not  agree  with 
the  law  of  chance.  Kiimelin,  in  his  Bevolkerungslehre, 
called  the  relatively  most  frequent  age  of  marriage 
the  ''  normal  "  age  of  marriage.  In  English  and  French 
works  it  is  quite  usual  to  call  the  relatively  most  frequent 
wage  rate  the  "  normal  rate  "  or  the  "  salaire  normal." 
In  French  writings  the  mode  is  called  by  some  authors 
simply  "  valeur  normale, "  or  briefly  "  normale."  This 
statistical  usage  agrees  with  the  common  usage.  If  non- 
statisticians  speak  of  a  normal  wage,  normal  income,  normal 
price,  etc.,  they  undoubtedly  think  of  the  relatively  most 
frequent  wage,  income  or  price.  The  same  average  is  in 
mind  when  we  speak  of  the  size  of  an  agricultural  state 
or  establishment  as  typical  for  a  certain  district. 

Special  importance  must  be  attributed  to  the  mode  as  a 
starting  point  for  measuring  the  dispersion  of  statistical 
series.  Only  in  series  distributed  symmetrically  around 
the  arithmetic  mean — and  in  this  case  arithmetic  mean 
and  mode  coincide — may  the  arithmetic  mean  be  used  as  a 
basis  for  measuring  the  dispersion  of  these  series.  How- 
ever, in  series  in  which  the  mode  does  not  coincide  with 
the  arithmetic  mean  we  may  also  start  from  the  mode  in 
order  to  examine  the  formation  of  the  parts  of  the  series 
on  both  sides  of  it.  If  in  such  a  case  we  start  from  the 
arithmetic  mean,  then  we  have  a  place  of  concentration  on 
one  side  of  it,  or,  in  graphic  representation,  a  ' '  hump  ' '  of 
the  curve,  to  which  no  similar  formation  corresponds  on 
the  other  side  of  the  arithmetic  mean.  However,  if  we 
start  from  the  mode  we  frequently  find  a  certain  regularity 
of  the  series.  The  most  pronounced  case  of  such  regularity 
is  given  when  the  series  coincides  with  a  skew  curve  of 
error  or  corresponds  to  the  asymmetric  Gaussian  law,  which 
is  the  ease  if  the  items  are  distributed  on  both  sides  of  the 


228  THE  VARIOUS  KINDS  OF  AVERAGES 

mode  according  to  the  normal  law  of  error  but  with  differ- 
ent degrees  of  precision  or  standard  deviation. 

The  above-mentioned  properties  of  the  mode  may,  in 
individual  cases,  appear  in  varying  degrees.  While  there 
are  series  which  possess  a  prominent  point  of  concentra- 
tion noticeable  at  the  first  glance,  around  which  the 
series  is  distributed  with  the  greatest  regularity,  on  the 
other  hand  we  sometimes  come  across  series  which  show 
only  indistinct,  hardly  ascertainable,  points  of  concentra- 
tion and  no  regular  grouping  around  these  at  all.  There- 
fore, the  scientific  value  of  the  mode,  its  applicability 
for  various  scientific  and  practical  purposes,  depends 
largely  on  the  formation  of  the  series  in  the  individual 
case. 

The  series  possessing  no  point  of  concentration  at  all,  as 
well  as  the  series  showing  two  and  more  points  of  con- 
centration, deserve  special  mention. 

Series  in  which  the  items  are  distributed  within  definite 
limits  without  any  point  of  concentration  whatsoever,  occur 
very  rarely.  Such  series  are  found,  however,  in  the  field 
of  anthropology.  Alphonse  Bertillon  has  based  his  anthro- 
pometric method  for  the  recognition  of  criminals  on  this 
fact.  If  certain  parts  of  the  bodies  of  a  great  number  of 
people  are  measured,  then  series  of  measurements  originate 
which  show  different  formations  according  to  the  parts 
of  the  body  measured.  The  measurements  of  certain  parts 
of  the  body,  as  experience  shows,  group  themselves  around 
a  relatively  most  frequent,  normal  measurement  (a  typical 
mean).  These  measurements  are  not  fit  for  the  identifica- 
tion of  individuals,  because  many  people  will  show  the  same 
value,  i.  e.,  the  most  frequent  measurement,  or  measure- 
ments approximating  this.  Therefore,  the  stature,  for 
instance,  possesses  only  small  "  signalizing  "  value  and 
plays  no  important  role  in  Bertillon 's  method  of  identifica- 
tion. Other  parts  of  the  body,  however,  have  no  typical 
value,  and  there  is  much  less  probability  that  the  same  meas- 


THE  MODE  229 

urement  occurs  with  many  different  people.  Objects  of 
measurement  which  produce  series  without  typical  mean 
are,  for  instance,  the  inside  length  of  the  leg,  the  width  of 
the  hips,  the  length  of  the  head,  the  reach  of  the  arms, 
the  length  of  the  foot,  and  the  length  of  the  middle 
finger.^^^ 

As  far  as  demographic  observations  in  the  precise  sense 
of  the  term  are  concerned,  it  seems  that  the  group  of  prema- 
ture deaths,  distinguished  by  Lexis  in  the  mortality  table, 
which  lies  between  the  group  of  childhood  mortality  and 
normal  old-age  mortality,  does  not  possess  a  place  of  con- 
centration.^^^  To  be  sure,  premature  deaths  cannot  be  re- 
produced in  an  independent  series,  since  the  deaths  cannot 
be  individually  assigned  to  the  three  groups  of  mortality 
mentioned.  However,  we  can  gain  an  idea  of  the  number 
and  distribution  of  the  premature  deaths,  by  producing  the 
curves  of  childhood  mortality  and  old-age  mortality  towards 
the  center  of  the  curve  which  represents  the  mortality  table, 
and  thus  deducting  a  number  of  deaths,  corresponding  to 
the  extreme  end  of  the  childhood  mortality  and  the  be- 
ginning of  the  old-age  mortality,  from  the  totality  of  the 
deaths  of  the  middle-age  classes.  The  remaining  premature 
deaths  are  distributed  rather  uniformly  over  the  middle-age 
classes  under  consideration  without  a  noticeable  point  of 
concentration. 

Series  having  two  or  more  modes,  and  consequently  re- 
sulting in  curves  with  two  or  more  culmination  points 

*^'With  a  height  of  from  1.60  to  1.65  meters  the  inside  length  of 
the  leg  varies  from  730  to  825  mm.  and  the  curve  of  distribution 
resulting  from  measurements  of  a  great  number  of  people  is  very 
long  and  irregular  (Lexis,  Abhandlungen,  VI,  "The  Typical  Values 
and  the  Law  of  Error,"  p.  125;  see  also  A.  Bertillon,  Das  anthro- 
pometrische  Signalement,  Lehrbuch  der  Identification,  German  by  v. 
Sury  [1895]). 

'^'  See  Lexis,  Zur  Theorie,  etc.,  p.  43,  and  Abhandlungen,  V,  "  On 
the  Causes  of  the  Small  Variability  of  Statistical  Relative  Xumbers," 
p.  88  f. 


230  THE  VARIOUS  KINDS  OF  AVERAGES 

when  reproduced  graphically,  occur  more  frequently  than 
series  without  any  mode.  The  modes  sometimes  stand 
equally  prominent,  sometimes  one  mode  is  most  prominent, 
but  other  modes  of  smaller  importance  (secondary  modes) 
may  be  established.  The  presence  of  several  modes  in  a 
series  usually  indicates  that  the  series  consists  of  hetero- 
geneous constituents,  the  typical  values  of  which  are  differ- 
ent from  each  other.  ^^^^ 

The  stature  of  the  population  of  some  districts  furnishes 
the  best  Imown  example  of  a  series  with  two  modes.  As 
early  as  1863  Adolphe  Bertillon  proved  the  occurrence  of 
several  modes  in  the  distribution  of  heights  of  the  recruits 
of  certain  French  Departments  ^^'^  and  the  demonstration 
was  substantiated  by  Jacques  Bertillon  in  1885,^""  and 
since  then  investigations  in  other  countries  have  led  to 
similar  results.  It  is  supposed  that  this  kind  of  distribu- 
tion of  heights  can  usually  be  explained  by  the  mixture 
of  several  races  of  different  typical  heights.  J.  Bertillon 
based  the  strange  distribution  of  height,  showing  two  modes, 
of  the  population  of  different  Departments  of  Northeastern 
France  on  the  fact  that  the  population  originated  from 
a  mixture  of  Celts  of  a  smaller  typical  height  and  Bur- 
gundians  of  a  larger  typical  height.  Quetelet  in  the  21st 
of  his  Letters  on  Prohdbilities  (Lettres  sur  les  proba- 
bilites)  had  already  pointed  to  the  fact  that  the  distribution 
of  heights  of  a  population  originating  from  the  mixture  of 
two  races  of  different  heights  would  indicate  the  fact  of 

*^*a  Sometimes  several  modes  are  caused  in  homogeneous  series 
of  measurements  because  of  concentration  of  the  items  at  customary 
values.  For  instance,  in  the  wage  data  of  the  Dewey  special  report 
on  Employees  and  Wages  (census  of  1900)  there  are  modes  at  the 
round  number  wage  points,  e.  g.,  $2.00  per  day  or  $2.50  per  day. 
Although  the  cause  of  such  modes  is  evident,  1.  e.,  accounting  con- 
venience, yet  there  is  no  reason  why  they  should  not  be  called  real 
modes. — Teakslatob. 

"'b  Bulletin  de  la  Societe  d'Anthropologie. 

^"cLa  Taille  de  I'homme  en  France. 


THE  MODE  231 

the  mixture  of  races,  although  he  then  had  no  material 
of  observation  at  hand."* 

Series  with  several  modes,  however,  are  found  not  only 
in  the  field  of  anthropometry.  Series  referring  to  social 
phenomena  often  show  several  modes,  especially  when  the 
totalities  represented  consist  of  several  heterogeneous  con- 
stituents. Thus,  the  duration  of  life  given  in  the  mortality 
table  shows  two  modes,  of  which  the  first  corresponds  to 
the  excessively  great  mortality  of  infants,  the  second  to 
the  "  normal  "  mortality  of  old  people.  Lexis,  evidently, 
was  induced  by  the  presence  of  these  two  modes  to  dis- 
tinguish three  groups  in  the  totality  of  deaths,  first,  the 
group  of  childhood  mortality,  second,  the  group  of  old-age 
mortality  (the  normal  group  of  deaths),  both  based  on 
pronounced  modes  (points  of  concentration),  third,  the 
group  of  premature  deaths,  which,  as  has  been  mentioned 
above,  are  distributed  without  pronounced  mode  over  the 
middle-age  classes  between  childhood  and  old  age. 

Other  series  of  age  data,  besides  the  mortality  table,  also 
often  have  two  or  more  modes  which  point  to  the  presence 
of  heterogeneous  constituents  within  the  totality  in  ques- 
tion. Thus,  the  age  constitution  of  persons  engaged  in 
certain  occupations  sometimes  shows  two  modes — for  in- 
stance, one  in  an  adolescent  class,  the  other  in  a  much 
higher  age  class.  Such  a  formation  of  the  series  points 
to  the  fact  that  principally  youthful  persons,  whose  capacity 
for  work  is  not  fully  developed,  and  persons  with  decreas- 
ing capacity  for  work  devote  themselves  to  that  occupation, 
while  strong  and  able  men  keep  away.     Also,  the  age  con- 

^^*  Not  only  different  nations  are  of  different  types  as  to  size,  but 
also  city  and  country  people  frequently  have  different  heights,  and 
also  the  height  for  different  occupations  is  different.  Therefore, 
the  constitution  of  a  population  of  city  and  country  people  and  of 
different  occupations  may  result  in  the  occurrence  of  several  culmina- 
tion points.  (See  Dr.  Siegfried  Rosenfeld,  "  Einige  Ergebnisse  aus 
den  Schweizer  Rekrutenuntersuchungen,"  AUg.  Stat.  Arcliiv,  Vol.  V, 
p.  124.) 


232  THE  VARIOUS  KINDS  OF  AVERAGES 

stitution  of  persons  convicted  of  certain  crimes  (as  arson, 
rape,  poisoning)  and  the  age  curve  of  certain  groups  of 
suicides  sometimes  have  several  modes,  Quetelet  noticed 
the  fact  that  principally  youths  and  old  men  are  guilty 
of  the  crime  of  rape.^^^ 

The  establishment  of  several  points  of  concentration  in 
a  series  may,  under  certain  circumstances,  induce  us  to 
form  "  natural  "  groups  which  correspond  to  the  various 
more  homogeneous  constituents  contained  in  the  total  series, 
rather  than  to  divide  the  series  into  classes  of  equal 
breadth.  A  series  showing  several  points  of  concentration 
ought,  in  such  cases,  to  be  divided  into  * '  natural  ' '  groups 
in  such  a  way  that  the  latter  coincide  as  much  as  possible 
with  the  constituents  distributed  around  the  various  points. 

The  establishment  of  several  points  of  concentration  in  a 
series  may  also  induce  us  to  separate  them  and  to  present 
the  heterogeneous  constituents  of  the  series  independently. 
But  this,  of  course,  presupposes  that  the  criterion  which 
is  the  basis  of  the  distinction  between  the  constituents  was 
included  in  the  observation  and  that  every  individual  case, 
when  observed,  was  examined  with  reference  to  this  cri- 
terion. If,  for  instance,  the  wages  of  all  the  workmen  of 
a  certain  district  are  ascertained  and  represented  jointly, 
then,  frequently,  a  series  with  several  points  of  concentra- 
tion is  found.  If  two  such  points  appear,  then  the  con- 
stituents which  are  distributed  around  them  may  be  dis- 
tinguished with  reference  to  the  sex  of  the  laborers. 
Women  usually  earn  lower  wages  than  men ;  therefore  the 
most  frequent  wage  for  women  does  not  coincide  with  the 
most  frequent  wage  for  men.  The  former  lies  in  a  lower 
wage  class,  a  fact  which  causes  two  points  of  concentration. 
If  the  sex  of  the  various  laborers  was  ascertained  in  the 
investigation,  then  it  is  easy  to  separate  men  and  women, 
i.  e.,  to  form  constituent  series  which  are  homogeneous  with 

""  tJber  den  Mensehen,  German  edition  by  Dr.  V.  A.  Rieeke,  1838, 
p.  547;  see  also  Ottingen,  Die  Moralstatistik,  3rd  ed.,  p.  519. 


THE  MODE  233 

reference  to  the  criterion  of  sex.  If  the  presence  of  two 
points  of  concentration  actually  was  the  consequence  of 
combining  the  sexes,  then  the  two  new  series  formed  will 
show  only  one  mode  each  and  a  more  regular  formation  in 
general.^"  Wage  data  for  workmen  of  only  one  sex  may 
also  have  several  points  of  concentration — for  instance,  if 
workmen  of  different  occupations  or  categories  of  work 
with  different  typical  wages  were  combined.  If  during  the 
investigation  the  occupations  of  the  individual  workmen  or 
the  different  categories  of  work  were  ascertained,  then  it  is 
possible  to  divide  the  totality  of  the  workmen  into  more 
homogeneous  constituents.  The  constituent  series  formed 
in  this  way  probably  will  have  only  one  mode  each,  around 
which  the  series  is  distributed  with  a  certain  regularity."^ 

2.   DETERMINATION  OP  THE  MODE 

The  determination  of  the  mode  of  a  series  of  individual 
observations  is  very  simple  if  the  series  shows  a  pronounced 
point  of  concentration  with  regular  distribution  of  the 
items  on  both  sides  of  it.  No  computation  at  all  is  re- 
quired ;  it  suffices  to  glance  over  the  series  to  ascertain  the 
mode.  In  graphic  reproduction  of  the  series  a  single, 
clear  culmination  point  is  seen,  the  abscissa  of  which  can, 
easily  be  read. 

"'Mortality  tables  which  have  been  computed  for  both  sexes, 
jointly,  also  show  two  points  of  concentration  of  old-age  mortality; 
if  the  two  sexes  are  separated,  more  regular  curves  are  found 
with  only  one  mode  each.  (See  Bulletin  de  I'lnstitut  international  de 
Statistique,  Vol.  X,  No.  1,  "  Vovimento  della  populazione  in  alcuni 
stati  d'Europa  e  d'America,"  Parte  II,  Statistica  delle  morti  negli 
anni  1874-1894.    Tavole  di  sopravivenza,  Tav.  I  f.,  Illf.). 

'^'  Numerous  illustrations  of  series  of  wage  data  which  show  sev- 
eral points  of  concentration  are  to  be  found  in  Bowley's  Elements 
of  Statistics.  Usually  (see  diagram  to  p.  145)  there  are  a  central 
mode  and  two  or  three  secondary  modes,  which  correspond  to  groups 
of  workmen  with  wages  either  far  above  or  far  below  the  average. 


234  THE  VARIOUS  KINDS  OF  AVERAGES 

However,  the  series  originating  from  statistical  observa- 
tions usually  are  not  of  an  entirely  regular  structure,  and, 
unless  the  statistician  is  working  on  the  original  material 
of  a  definite  statistical  investigation,  he,  in  most  cases, 
does  not  have  to  do  with  individual  observations  given  as 
such.  On  the  contrary,  he  has  to  work  with  series  that 
consist  of  classes  and  merely  show  the  number  of  observa- 
tions belonging  to  the  different  classes.  Now  this  fact  is 
not  at  all  unfavorable  to  the  determination  of  the  mode,  as 
we  might,  perhaps,  think  at  first.  As  has  been  mentioned, 
the  individual  items  of  a  statistical  series  hardly  ever 
show  an  entirely  regular  structure,  and,  in  the  compass 
of  the  relatively  most  frequent  items,  very  irregular  fluctua- 
tions are  found  which  render  it  impossible  to  determine  the 
exact  mode  on  the  basis  of  the  individual  items.  Only  by 
the  formation  of  classes,  in  which  the  accidental  fluctuations 
counterbalance  each  other,  does  the  series  receive  a  more 
regular  structure  which  reveals  the  real  points  of  con- 
centration. Therefore,  if  a  series,  the  mode  of  which  is  to 
be  determined,  does  not  consist  of  classes  from  the  be- 
ginning, but  shows  the  original  individual  data,  then  it 
is  expedient  in  most  cases  to  first  condense  the  series  in 
well  chosen  classes.  To  be  sure,  the  result  of  this  procedure 
is  the  further  task,  sometimes  connected  with  difficulties, 
of  determining  the  exact  position  of  the  mode  in  the  class 
of  the  greatest  frequency. 

A  strange  statistical  phenomenon  is  found  in  the  fact, 
observed  especially  by  Bowley,  that  the  position  of  the 
mode  is  dependent  on  the  nature  (breadth  and  position) 
of  the  classes  of  which  the  series  consists.  If  we  condense 
the  same  original  material  several  times  into  classes  of 
different  breadths,  or  if  we  form  classes  of  the  same  breadth 
but  of  different  limits,  then  it  may  happen  that  the  series 
thus  formed  have  different  points  of  concentration.  There- 
fore, in  order  to  ascertain  the  correct  mode,  it  is  sometimes 
expedient   to   condense    the    same    items   repeatedly    into 


THE  MODE  235 

classes  of  varj'ing  breadths  and  positions  in  order  to  find 
by  experimenting  (methode  de  tatonnement,  Liesse)  that 
kind  of  group  formation  which  is  best  suited  to  the  struc- 
ture of  the  totality  and,  consequently,  represents  the  posi- 
tion of  the  mode  most  correctly.  The  arithmetic  mean  and 
the  median  can  be  computed  the  more  accurately,  the  more 
detailed  are  the  classes  in  which  the  items  are  distributed. 
However,  this  does  not  hold  for  the  computation  of  the 
mode.  Classes  of  greater  breadths  may  show  the  mode 
more  plainly  than  more  detailed  groups  of  lesser  breadths. 
In  illustration  of  this  Bowley  gives  in  his  Elements  of 
Statistics  (p.  119)  the  following  example,  which,  it  is  true, 
is  based  on  a  very  irregular  series,  so  that  the  computation 
of  the  mode  is  connected  with  special  difficulties. 

The  point  in  question  is  to  ascertain  the  mode  of  a  series 
taken  from  the  U.  S.  Report  on  ^Yholesale  Prices,  Wages 
and  Transportation  (1893).  Bowley  tabulates  the  wages 
of  5,123  workmen.^^^  He  combines  the  items  successively 
into  classes  of  10  cents,  20  cents,  30  cents,  and  50  cents  in 
breadth."^  In  the  series  of  10-cent  grouping  the  wage  class 
$1.15-$1.24  has  the  greatest  frequency;  to  it  belong  685 
workmen.  However,  the  series  of  10-cent  groupings  is,  as 
Bowley  explains,  of  very  irregular  structure  ;  the  series  con- 
tains, strictly  speaking,  14  points  of  concentration,  of  which, 
however,  the  one  in  the  wage  class  $1.15-$1.24  is  the  most 
prominent.  In  the  series  of  20-cent  classes  the  numbers 
of  workmen  belonging  to  the  different  classes,  starting  the 
lowest  class  at  25  cents,  are:  16,  144,  270,  370,  989,  557, 
538,  531,  etc.  The  number  989  for  the  wage  class  $1.05- 
$1.24  means  a  pronounced  mode.  If  we  start  the  lowest 
class  at  35  cents,  w^e  obtain  the  numbers :  74,  242,  282,  505, 
784,  924,  274,  etc.     The  mode  lies  in  the  wage  class  $1.35- 

"'This  is  the  same  series  of  wage  data  which  Bowley  has  used 
also  in  the  graphical  determination  of  the  median.  (See  above, 
p.  214). 

'•"See  the  tables  (p.  91  and  p.  120)  in  Elements  of  Statistics. 


236  THE  VARIOUS  KINDS  OF  AVERAGES 

$1.54,  to  which  924  workmen  belong.  Thus  two  series,  both 
of  which  contain  classes  of  20  cents  breadth  each,  show 
the  mode  in  quite  different  wage  classes.  This  shows  that 
the  mode  cannot  be  ascertained  at  all  by  forming  classes 
of  this  breadth.  If  30-cent  classes  are  formed,  then  we 
obtain,  according  as  we  start  from  55  cents,  65  cents,  or 
75  cents  when  forming  the  groups,  the  following  values: 
355,  674,  1,242  (for  the  wage  class  $1.15-$1.44),  740,  etc., 
or  439,  1,190  (for  the  wage  class  $0.95-$1.24),  1,023,  etc., 
or  483,  1,088  (for  the  wage  class  $1.05-$1.34),  996,  etc. 
The  wage  classes  of  the  greatest  frequencies  in  the  three 
series  are:  $1.15-$1.44;  $0.95-$1.24;  and  $1.05-$1.34.  The 
three  wage  classes  partly  coincide,  inasmuch  as  all  three 
contain  the  narrower  wage  class  $1.15-$1.24.  Therefore, 
we  may  assume  that  this  wage  class  contains  the  mode.  It 
would  lie,  perhaps,  in  the  middle  of  this  wage  class,  and 
thus  would  be  about  $1.20. 

Bowley  summarizes  the  method  of  computing  the  mode 
described  above  as  follows:  "  Tabulate  the  figures  again 
and  again  in  gradually  widening  groups  till  regularity  is 
obtained;  then  examine  again  the  groups  which  have  the 
selected  width  and  see  if  the  mode  is  shifted  when  the 
lower  limit  of  the  grouping  is  moved;  if  it  is  shifted  the 
groups  are  not  wide  enough;  if  it  is  not,  the  mode  is  in 
the  smallest  group  common  to  the  larger  equal  groups 
which  all  contain  it."^-**  Of  course  only  statistical  bu- 
reaus having  the  original  material  at  hand  are  able  to  follow 
Bowley 's  method.  These  only  are  able  to  form  experimen- 
tal classes  of  any  breadth  and  position.  The  statistician 
who  works  on  a  series  contained  in  a  statistical  publication 
can  form  groups  of  greater  breadth  merely  by  adding  ad- 
joining classes,  but  he  cannot  divide  the  given  classes  nor 
change  their  limits  except  by  means  of  hypothetical  in- 
terpolation. 

Having  found  the  class  in  which  the  mode  of  the  items 

^=»  Ibid.  p.  121. 


THE  MODE  237 

lies,  we  can  determine  the  mode  more  accurately  within  the 
limits  of  this  elass.^^^  In  the  illustration  quoted  above 
(see  p.  236)  Bowley  has  assumed  that  the  mode  lies  in  the 
center  of  the  class  of  the  greatest  frequency.  This  assump- 
tion corresponds  to  the  hypothesis  that  the  items  within 
the  limits  of  the  class  of  the  greatest  frequency  are  sym- 
metrically distributed  around  the  mode.  But  the  struc- 
ture of  the  series  may  be  such  that  the  items  may  have 
an  asymmetrical  distribution  within  the  limits  of  the  class 
of  the  greatest  frequency.  Therefore,  in  such  a  case,  an- 
other hypothesis,  more  suited  to  the  structure  of  the  series, 
may  be  chosen.  We  may  also  resort  to  graphic,  or,  if  we 
want  to  proceed  with  very  great  accuracy,  to  algebraic 
interpolation.  Bowley  has  found  the  mode  in  the  series 
of  wages  of  the  American  workmen,  by  means  of  algebraic 
interpolation,  to  be  $1.10,^^2  while  the  elementary  method 
gave  $1.20. 

As  is  Imown,  statistical  series  are  frequently  subjected 
to  adjustment  in  order  to  make  their  characteristic  forma- 
tions appear  more  clearly  and  to  remove  the  accidental 
irregularities  in  their  structures.  The  adjustment  may 
affect  the  numerical  value  of  the  mode.  The  mode  of  the 
adjusted  series  or  (in  case  of  graphic  adjustment)  of  the 
adjusted  curve  may  be  different  from  the  mode  of  the  un- 
adjusted series  or  the  unadjusted  broken  line.  Since  the 
series  receives  a  more  regular  formation  by  adjustment, 
the  mode  of  an  adjusted  series  is  more  clearly  seen  and 
can  be  computed  with  greater  accuracy.  Since  adjustment 
removes  accidental  irregularities,  the  mode  of  an  adjusted 
series  possesses  greater  theoretical  value  than  the  mode 
ascertained  from  the  original  material  of  the  series.  It 
approaches  that  ideal  mode  which  we  would  obtain  from 
an  infinitely  large  number  of  observations  with  infinitesimal 

*''  In  tabular  presentation  of  series  the  class  of  greatest  frequency 
is  usually  set  off  by  heaA^  print  or  print  of  different  color. 
^"  Elements  of  Statistics,  2nd  ed.,  p.  254. 


238  THE  VARIOUS  KINDS  OF  AVERAGES 

differences  between  the  individual  measurements.  If  the 
adjustment  is  done  by  means  of  an  analytical  formula,  then 
the  mode  may  also  be  computed  directly  from  it  as  the 
maximum  of  the  function  used.^^^ 

A  peculiar  graphic  method  of  computing  the  mode  is 
based  on  that  kind  of  group  formation,  advocated  especially 
by  Galton,  in  which  we  are  not  shown  how  many  cases 
belong  to  each  class,  but  what  numerical  sums  we  obtain, 
if,  starting  from  the  lowest  or  highest  class,  we  add  the 
single  classes  successively  and  form  so-called  "  cumulative 
classes.  "^^*  Bowley  explains  this  method  of  computing 
the  mode  in  his  Elements  of  Statistics  ^^^  on  the  basis  of 
the  same  American  wage  data  which  he  has  also  used  in  the 
graphic  determination  of  the  median  and  which  have 
served  above  (p.  235  f.)  to  illustrate  the  dependence  of  the 
mode  on  the  kind  of  group  formation.  These  wage  data 
condensed  into  "  cumulative  "  classes  are  reproduced 
graphically  by  Bowley  so  that  the  abscissas  of  the  single 
points  of  the  diagram  correspond  to  definite  wage  grades 
while  the  ordinates  indicate  the  numbers  of  Avorkmen  earn- 
ing at  or  above  a  certain  wage.  The  line  obtained  in  this 
way  ascends  throughout  from  right  to  left,  since  the  axis 
of  the  abscissas  indicates  the  different  wage  grades  from 
right  to  left  in  decreasing  order  and  the  number  of  work- 
men earning  at  or  above  a  certain  wage,  naturally,  is  the 
greater,  the  lower  the  wage. 

The  mode  is  indicated  in  this  diagram  by  that  point  at 
which  the  greatest  number  of  workmen  is  added;  it  is 

»^'  See  especially  Fechner,  Kollektivmasslehre,  pp.  183-186,  and 
Lucien  March,  "  Quelques  examples  de  distribution  des  salaires," 
Journal  de  la  Societe  de  Statistique  de  Paris,  1898,  pp.  199  f.,  205, 
and  247. 

^=*  Cf.  details  about  this  method  of  group  formation,  p.  85  f.; 
and  about  the  use  of  such  cumulative  classes  in  the  graphic  de- 
termination of  the  median,  p.  213  f. 

""  2nd  ed.,  p.  155. 


THE  MODE  239 

that  place  where  the  diagram  is  steepest.  After  we  have 
adjusted  the  diagram  and  drawn  a  curve  instead  of  the 
broken  line,  the  tangent  crosses  the  curve  at  this  place. 
The  point  of  crossing  can  be  found  mechanically  by  placing 
a  ruler  to  the  curve  and  turning  it  until  it  crosses  the 
curve.  The  mode  and  the  secondary  points  of  concentra- 
tion can  also  be  recognized  by  the  fact  that  at  the  points 
in  question  the  curve  changes  from  concavity  toward  the 
axis  of  abscissas  into  convexity.  Bowley's  curve  is  con- 
cave to  the  base-line  from  $0.30  to  about  $1.20,  convex 
from  about  $1.20  to  about  $3.15,  then  again  concave  till 
$3.40,  and  then  convex  till  its  right  end.  Thus  it  has  two 
points  of  concentration,  one  at  about  $1.20,  the  second, 
which  is  of  much  less  importance,  at  about  $3.40.  The 
investigation  of  the  numerical  material,  as  carried  out 
above,^^''  has  also  placed  the  mode  at  $1.20.  By  means  of 
algebraic  interpolation  Bowley  has  found  the  two  modes  to 
be  $1.10  and  $3.20.^^^  These  divergences  again  show  how 
difficult  it  is  to  determine  accurately  the  position  of  the 
mode  and  of  the  secondary  points  of  concentration. 

Bowley  advocates  the  above  graphical  method  of  de- 
termination because  the  computation  of  the  mode  from  the 
numerical  material  of  the  series  meets  with  difficulties  on 
account  of  the  unequal  distribution  of  the  items  on  both 
sides  of  the  mode  and  on  account  of  the  dependence  of 
the  numerical  size  of  the  mode  on  the  kind  of  group  forma- 
tion chosen  in  the  individual  case.  He  explains  that,  when 
this  graphic  method  is  used,  the  first  of  these  difficulties  is 
removed  entirely  and  the  second  is  decreased,  since  with 
the  use  of  this  method  only  unessential  changes  of  the  mode 
according  to  the  kind  of  adjustment  of  the  diagram  can 
occur.  Another  advantage  which  Bowley  ^^^^  attributes  to 
this  graphic  method  of  computing  the  modes,  is  the  follow- 
ing: "  This  method  can  be  applied  to  numbers  which  are 
given  at  irregular  intervals  of  graduation  (e.  g.,  30  at  30  s. 

"«P.  236  f.  ^"Elements,  p.  254.  "'a  Elemeuts,  p.  156. 


240  THE  VARIOUS  KINDS  OF  AVERAGES 

6  d.,  40  at  30  s.  81/0  d.,  35  at  40  s.  1  d.,  etc.)  as  easily  and  by 
exactly  the  same  construction  as  to  more  regular  returns; 
and  if  the  smooth  curve  is  regularly  drawn,  the  number  of 
modes  can  be  seen  at  a  glance  and  the  individual  importance 
of  each  can  be  estimated. ' ' 


3.   APPLICATIONS  OP  THE  MODE 

The  determination  of  the  mode  has  become  customary 
in  many  fields  of  statistics.  To  be  sure,  the  mode,  as  has 
already  been  mentioned,  is  frequently  not  determined  ex- 
actly, but  only  the  class  of  the  greatest  frequency  is  em- 
phasized. 

In  the  field  of  population  statistics  series  of  age  data 
frequently  suggest  the  computation  of  the  mode  which 
indicates  the  "  normal  age  "  of  the  group  of  population  in 
question.  Thus,  the  mode  is  often  computed  from  the 
age  constitution  of  those  marrying  and  is  called  the  "  nor- 
mal age  of  marriage. ' '  This  normal  age  of  marriage  obvi- 
ously varies  for  each  of  the  two  sexes  and  also  for  different 
social  groups. 

One  of  the  most  interesting  applications  of  the  mode  is 
the  **  normal  length  of  life  "  (called  also  "  normal  life- 
time," "  normal  age  "),  a  notion  which  Lexis  has  intro- 
duced with  great  success  into  statistical  literature.^^^  The 
normal  length  of  life  is  the  mode  of  the  series  of  lifetimes 
contained  in  the  mortality  table,  it  is  that  age  at  which — 
with  the  exception  of  infancy — most  people  die  according 

*'°  Lexis  has  fully  explained  the  term,  normal  length  of  life,  in 
several  of  his  writings  and  advocated  its  use  in  the  Paris  demo- 
graphical  congress  of  1878  (see  Zur  Theorie,  etc.,  1877,  pp.  42- 
64;  Annales  de  demographie  Internationale,  1878,  p.  447;  1880,  p. 
481,  and  1881,  p.  233:  "  Sur  les  moyennes  appliquees  aux  mouve- 
ments  de  la  population  et  sur  la  vie  normale " ;  Abhandlungen,  VI, 
"The  Typical  Values  and  the  Law  of  Error,"  pp.  10-119.  See  also 
Czuber,  Wahrscheinlichkeitsrechung,  p.  337  ff.). 


THE  MODE  241 

to  the  mortality  table  and  may,  therefore,  be  considered 
to  be  the  normal,  typical  length  of  life. 

The  normal  length  of  life  in  Germany  is  71  years  for 
both  sexes  combined.  In  most  of  the  other  countries  it 
also  lies  at  the  beginning  of  the  seventies.  The  normal 
length  of  life  of  women  usually  is  considerably  greater  than 
that  of  men.  In  Germany  the  difference  is  about  two 
years.^^* 

It  is  especially  interesting  that  the  deaths  above  the 
normal  age  and  those  in  the  age  classes  next  below  the 
normal  age — the  deaths  of  the  normal  group  of  mortality 
distinguished  by  Lexis — are  usually  distributed  symmetric- 
ally and  according  to  the  law  of  error  about  their  mode. 
Therefore,  the  latter  appears  to  be  a  typical  mean  in  the 
strict  mathematical  sense,  if  not  with  regard  to  the  whole 
mortality  table,  at  least  with  regard  to  a  considerable  part 
of  it,  i.  e.,  with  regard  to  the  group  of  normal  mortality 
of  old  age.  Indeed,  the  normal  length  of  life  is  the  only 
average  of  a  typical  character  which  can  be  obtained  from 
the  mortality  table.  Mean  and  probable  length  of  life, 
which  are  based  upon  all  the  values  contained  in  the 
mortality  table,  are  non-typical,  i.  e.,  they  fall  in  age 
classes  which  are  of  low  frequency  in  the  series  of  life- 
times. This  is  caused  by  the  facts,  that  the  totality  of 
deaths  consists  of  the  three  groups  of  childhood  mortality, 
premature  deaths,  and  normal  old-age  mortality,  as  dis- 
tinguished by  Lexis,  and  that  masses  which  are  composed 

'"  For  Austria  two  ungraduated  mortality  tables  are  given,  com- 
puted according  to  somewhat  different  methods  on  the  basis  of  the 
results  of  the  census  of  December  31,  1900.  (See  Die  Ergebnisse  der 
Volksziihlung  vom  31  Dez.,  1900,  Osterr.  Statistik,  Vol.  XLV,  No.  5, 
supplement.)  The  computation  of  one  of  these  tables  is  based  on 
the  average  mortality  of  the  six  years  preceding  the  census;  the 
computation  of  the  other  is  based  on  the  mortality  in  the  two  years 
before  and  after  the  census  year.  The  former  table  of  mortality 
shows  the  largest  number  of  deaths  for  both  sexes  in  the  71st  year  of 
age,  the  latter  also  for  both  sexes  at  the  end  of  the  70th  year. 


242  THE  VARIOUS  KINDS  OF  AVERAGES 

of  heterogeneous  constituents  are  not  regularly  distributed 
as  a  whole  around  a  "  typical  "  mean. 

Besides  the  age  constitution  of  the  mortality  table,  the 
age  constitution  of  the  deceased  usually  contains  a  mode 
corresponding  to  the  typical  old  age.  But  just  as  the  aver- 
age age  of  the  deceased  is  of  lesser  scientific  value  than 
expectation  of  life  computed  from  the  mortality  table,  the 
mode  in  the  age  constitution  of  the  deceased  is  of  less 
value  than  the  corresponding  mode  in  the  series  of  life- 
times of  the  mortality  table. 

The  determination  of  the  mode  is  of  considerable  im- 
portance in  the  field  of  wage  statistics.  Evidently  it  is  of 
great  importance  to  know  what  wages  the  relatively  greatest 
number  of  workmen  earns.  This  wage  can  be  considered 
to  be  the  normal,  typical  wage.  As  a  matter  of  fact  the 
most  frequent  wage  is  usually  computed  in  modern  works 
on  wage  statistics  that  are  based  on  individual  wage  data — 
for  instance,  in  the  Austrian  wage  statistics  for  the  miners 
of  the  Ostrau-Karwin  coal  district.^^"  Bowley  has  illus- 
trated his  methods  of  the  computation  of  the  mode  by 
use  of  wage  statistics.  In  his  works  in  the  field  of  his- 
torical wage  statistics  Bowley  has  tried  especially  to  estab- 
lish the  fluctuation  of  the  "  normal  wage  rates  "  (pre- 
dominant rates)  of  the  English  workmen  in  the  course 
of  the  19th  century.  Therefore  he  has  used,  as  much  as 
possible,  the  lists  of  '*  majority  rates  "  published  by  the 
employers'  associations,  which  state  the  wage  rates  on  the 
basis  of  which  relatively  most  workmen  are  paid  in  the 

*'°  See  Arbeiterverhaltnisse  im  Ostrau-Karwiner  Steinkohlenreviere, 
k.  k.  Arbeitsstatischen  Amte  im  Handelsministerium,  1st  part, 
Vienna,  1904,  p.  29  ff.  In  Table  V  of  the  publication  mentioned  the 
miners  investigated  are  divided  into  wage  classes  according  to  the 
size  of  their  wages,  and  the  classes  are  designated  to  which  relatively 
most  workmen  of  the  different  categories  of  workmen  belong.  See 
also  p.  60;  there  the  strongest  groups  are  given  on  the  basis  of 
Table  IX,  in  which  the  miners  are  divided  into  groups  according  to 
their  annual  incomes  from  their  work. 


THE  MODE  243 

various  oceupations.^^^-^^^-^^-*  In  modem  Italian  statistics, 
too,  the  relatively  most  frequent  wages  (i  salari  piu  fre- 
quenti )  of  definite  categories  of  workmen  are  frequently  de- 
termined. ^^^ 

Estimated  modes  have  an  extensive  use  in  the  field  of 
wage  statistics.  Frequently,  not  series  of  individual  wage 
data  are  procured — from  which  the  mode  can  be  computed 
afterwards — but  in  many  instances  the  mode  itself  forms 
the  primary  object  of  the  question,  and  this  compels  the 
person  asked  to  estimate  the  mode.  Thus,  experts  are 
frequently  questioned  as  to  the  relatively  most  frequent, 
normal  wage  of  the  workmen  of  their  district.  Since  they 
have  no  individual  data  at  their  disposal,  they  have  to 
resort  to  estimates.^^* 

"' Bowley  and  Wood,  "The  Statistics  of  Wages  in  the  United 
Kingdom  during  the  Nineteenth  Century,"  Pt.  XIV,  Journal  of  the 
Royal  Statistical  Society,  Vol.  LXIX  (1906),  Pt.  I,  p.  15o. 

"^  Also  Prof.  Mandello  has  used  the  "modus"  in  his  works  in 
the  field  of  historical  wage  statistics  (published  in  the  Hungarian 
language)  ;  see  the  Bulletin  de  I'lnstitut  international  de  Statistique, 
Vol.  XIII,  No.  1,  p.  401. 

132a  The  mode  has  also  been  used  recently  by  the  English  Board 
of  Trade  in  their  investigations  of  the  wages  and  hours  of  labor, 
rents  and  housing  conditions,  retail  prices  of  food  and  the  expendi- 
ture of  working-class  families  on  food  in  the  more  important  in- 
dustrial towns  of  important  countries.  Reports  have  been  issued 
for  the  United  Kingdom  (Cd.  3864),  Germany  (Cd.  4032),  France 
(Cd.  4512),  Belgium  (Cd.  5065),  and  the  United  States  (Cd.  5609). 
— Translator. 

^"  See  "I  salari  e  gli  orari  piQ  frequenti  per  gli  operai  orga- 
nizzati  in  Genova,"  Bolletino  dell'  Ufficio  del  lavoro.  May,  1905, 
p.  735. 

"*  In  the  investigation,  already  mentioned,  of  the  conditions  of 
workmen  in  the  Ostrau-Karwin  coal  district,  which  the  Austrian 
Labor  Statistical  Bureau  made  in  1901,  the  individual  earnings  of 
the  miners  were  ascertained;  but  for  purposes  of  comparison,  data 
on  the  conditions  of  workmen  in  small  industrial  establishments  in 
the  same  district  were  also  obtained,  by  means  of  an  interrogatory 
concerning  the  normal  wage  per  week  for  workmen  of  various  Indus- 


244  THE  VARIOUS  KINDS  OF  AVERAGES 

Also,  individual  workmen  are  sometimes  asked  about 
their  normal  wage,  for  instance,  per  week  during  the  last 
year.  They  would  be  able  to  compute  the  relatively  most 
frequent  weekly  wage  from  the  *'  series  "  of  the  weekly 
wages  which  they  have  received  in  the  course  of  that  year, 
provided  they  have  kept  a  record  of  the  single  weekly 
payments.  But  this  is  not  often  the  ease.  Therefore  the 
workmen  usually  will  make  a  more  or  less  arbitrary  esti- 
mate of  their  "  normal  "  weekly  wage. 

The  relatively  most  frequent  working  time  deserves  the 
same  consideration  in  the  presentation  of  the  conditions 
of  working  hours  as  does  the  relatively  most  frequent  wage 
in  the  presentation  of  wage  conditions.  Figures  giving 
the  relatively  most  frequent  working  time  in  specified  occu- 
pations are  to  be  found,  for  example,  in  the  bulletins  of 
the  Italian  labor  bureau.^^^  Also  in  other  publications 
the  relatively  most  frequent  working  time  is  frequently 
taken  into  due  consideration.^^^ 

In  price  statistical  publications  the  price  is  frequently 
stated  at  which  the  largest  number  of  units  of  a  commodity 
are  sold.  Presumably  this  price  is  meant  when  the 
"  usual  "  local  price  of  a  commodity  is  computed  from 
detailed  data  or  is  asked  for  directly .^^^    Also  in  the  field 

tries  and  categories  for  the  year  1901.  (See  Arbeiterverhaltnisse 
im  Ostrau-Karwiner  Steinkohlenreviere,  1st  part,  Vienna,  1904,  p.  li.) 

*'*  Cf.  in  the  May  number,  1905,  p.  735,  "  I  salari  e  gli  orari  piil 
frequenti   per  gli   operai   organizzati   in   Genova." 

^'»  Thus  in  the  "  Statistik  der  Stadt  Zurich,"  No.  1  (published  by 
the  Statistical  Bureau  of  the  City  of  Zurich  in  1904),  we  learn  that 
the  net  working  time  of  the  workmen  in  municipal  service  is  8V2  to 
12,  most  frequently  ten  hours,  in  summer  time,  and  7%  to  12V2>  most 
frequently  nine  hours,  in  winter  time.  (See  Soziale  Evmdschau 
[Vienna],  1905,  Pt.  I,  p.  5.) 

^°'  In  the  investigation  of  the  Austrian  Bureau  of  Labor  Sta- 
tistics in  1901  concerning  the  conditions  of  the  miners  in  the 
Ostrau-Karwin  coal  districts  there  was  also  used  an  interrogatory 
to  ascertain  the  local  retail  prices  of  certain  commodities.    The  usual 


THE  MODE  245 

of  historical  price  statistics  the  mode  has  been  used,  though 
rarely,  for  the  computation  of  total  index  numbers,  i.  e.,  of 
averages  of  the  indices  which  represent  the  price  fluctua- 
tions of  certain  commodities  for  a  number  of  years. 

Finally,  we  may  point  to  the  frequent  use  of  the  mode 
in  anthropological  statistics.  Series  of  anthropometric 
data,  as  a  rule,  contain  one  or  more  ^^^  visible  points  of 
concentration  which  must  be  especially  emphasized.  How- 
ever, there  also  occur  anthropometric  series  without  a 
mode,  as  has  been  mentioned  above  (on  p.  228). 

The  mode  is  of  special  importance,  since  it  is  that  aver- 
age which  is  easiest  to  estimate  and,  therefore,  can  easiest 
be  obtained  in  an  investigation  by  direct  questioning.  It 
has  been  mentioned  that  in  investigations  the  questions 
asked  are  frequently  for  the  "  predominant,"  "  prevail- 
ing," "  normal,"  or  "  usual  "  price  or  wage,  and  these 
questions  are  asked  of  persons  who  are  considered  to 
be  especially  competent  to  correctly  estimate  the  "  nor- 
mal "  wage  and  the  "  usual  "  price  on  the  basis  of 
their  experience.  The  normal  or  usual  magnitude  is  often 
asked  for  because  it  is  much  easier  for  an  expert 
to  estimate  the  mode  than  any  other  average.  To 
determine  a  geometric  or  an  arithmetric  mean  it  is  neces- 
sary to  know  all  the  individual  cases,  since  these  must  be 
added  or  multiplied.  Also,  to  determine  the  median  it  is 
necessary  to  be  able  to  survey  the  sizes  of  the  individual 
cases  enough  to  find  the  item  which  lies  in  the  center  of 
the  series  arranged  according  to  the  sizes  of  the  items. 
The  computation  of  the  means  mentioned  (arithmetic  mean, 
geometric  mean,  and  median)  therefore  presupposes  that 
the  items,  the  mean  of  which  is  to  be  computed,  are  known. 

prices  were  asked  for,  and  in  case  of  great  fluctuations  of  price 
also  the  lowest  and  highest  prices.  (Cf.  Arbeiterverhaltnisse  im 
Ostrau-Karwiner  Steinkohlenreviere,  Pt.  I,  p.  xliii.) 

^''  See  above  on  anthropometric  series  with  two  points  of  con- 
centration, p.  229. 


246  THE  VARIOUS  KINDS  OF  AVERAGES 

If  this  is  the  case  these  means  can  be  computed  correctly. 
If,  however,  the  items  are  not  known — and  we  are  con- 
sidering this  case — then  the  means  mentioned  cannot  be 
estimated  at  all  or  only  with  great  difficulty.  None  of  these 
means  may  have  actually  made  its  appearance  in  a 
definite  individual  case,  and  the  person  asked  is  usually 
not  able  to  give  any  information  about  any  of  these  means 
on  the  basis  of  his  own  perception,  on  the  basis  of  his 
memory.  Therefore,  direct  estimation  would  have  to  be 
quite  arbitrary  on  account  of  the  lack  of  information.  In 
order  to  get  at  a  value  in  an  indirect  way  the  person  asked 
would  have  to  estimate  the  sizes  of  all  the  individual  cases 
to  be  taken  into  consideration  and  then  compute  the  arith- 
metic or  geometric  mean  or  the  median  from  these  esti- 
mated items,  a  procedure  which  would  be  much  too  intricate 
for  practical  purposes. 

The  relatively  most  frequent  value  is  quite  different. 
The  relatively  most  frequent,  or  "  normal,"  size  of  a 
phenomenon  impresses  itself,  even  upon  the  passive  ob- 
server, as  a  fact  of  experience,  and  every  expert  is  able  to 
state  from  memory  and  without  further  computation  that 
size  which  he  has  perceived  most  often.  Therefore,  if  in- 
dividual observations  (individual  data)  are  not  given  and 
if  the  mean  size  of  a  phenomenon  is  to  be  estimated,  we 
must  ask  for  the  relatively  most  frequent  size  in  a  theoret- 
ically correct  way,  because  only  this  average  admits  of  direct 
estimation.^^^ 

*'•  In  this  sense  the  questionnaire  planned  by  Marcus  Rubin  for 
demographic  observations  in  countries  without  a  census,  and  meant 
to  be  answered  by  travelers,  missionaries,  etc.,  contains  the  following 
questions : 

A  quel  age  se  marie-t-on  [ordinairement?  les  hommes?  lea 
femmes?],      (Question  27.) 

Pent  on  estimer  le  nombre  d'enfants  qu'a  ordinairement  una  fipousef 
(Question  28.) 

Sur  les  explorations  dfimographiques  h  exficuter  dans  les  pays  oil 


THE  MODE  247 

However,  this  postulate  is  not  always  taken  into  account. 
Frequently  we  ask  for  an  estimate  of  the  arithmetic  mean 
of  definite  phenomena.  But  the  different  kinds  of  means 
are  not  always  sufficiently  distinguished  in  practice.  The 
persons  intrusted  with  the  estimation  of  an  arithmetic 
mean  often  choose  that  mean  which  is  easiest  to  compute. 
"With  estimated  means,  it  is,  therefore,  often  doubtful 
whether  they  actually  correspond  to  the  arithmetic  mean  or 
to  the  mode. 

The  fact  that  of  all  means  the  mode  is  the  easiest  to 
estimate  makes  it  the  most  widely  used  mean  of  everyday 
life.  If  the  man  in  the  street  wants  to  characterize  a 
variable  phenomenon  by  a  single  expression  he  usually  re- 
sorts to  the  relatively  most  frequent  size  which  has  clung 
to  his  memory,  and  he  feels  instinctively  that  this  value  has 
a  special  importance,  that  it  indicates,  so  to  speak,  the  normal 
case  of  the  phenomenon.  And  while  doing  this  even  less 
educated  persons  are  usually  conscious  of  the  fact  that  the 
relatively  most  frequent  case,  which  they  are  using  to 
characterize  a  phenomenon,  is  not  identical  with  its  arith- 
metic mean.  The  author  has  made  numerous  psycholog- 
ical experiments  concerning  this  point.  He  has  asked  con- 
ductors of  tramways,  omnibuses,  etc.,  how  long  they  average 
for  covering  certain  distances  and,  frequently,  has  received 
the  answer  that  the  conductor  could  not  say  exactly,  that 
he  needed  * '  sometimes  longer,  sometimes  shorter,  usually  so 
and  so  many  minutes."  If  we  ask  different  people  at 
what  time  they  get  up,  go  to  sleep,  or  similar  question,  we 
frequently  receive  the  answer,  that  they  could  not  state  the 
exact  time  when  this  takes  place,  but  usually  at  such  and 
such  a  time. 

il  n'existe  pas  encore  de  recensement.     (Report  of  the  9th  session  of 
the  Intern.  Stat.  Inst,  in  Berlin,  1903.) 


PART  III 

DISPEESION  ABOUT  THE  MEAN 
OK  AVERAGE 


CHAPTER  I 

PURPOSE  AND  MEANING  OF  ESTABLISHING  THE  DIS- 
PERSION OF  STATISTICAL  SERIES 

When  investigating  the  dispersion  (grouping,  distribu- 
tion, spreading)  of  the  items  of  a  statistical  series  around 
their  mean  either  of  two  different  aims  may  be  pursued. 
The  object  may  be  closer  characterization  of  the  mean  or  a 
closer  characterization  of  the  series. 

Every  average  gives  certain  information  about  the  series 
from  which  it  has  been  computed,  but  it  does  not  express 
the  formation  of  the  series.  Series  very  differently  con- 
stituted may  result  in  means  of  the  same  numerical  size. 
This  is  the  reason  why  so  many  people  are  directly  opposed 
to  the  use  of  means.  Numerous  statisticians  prefer  the 
presentation  of  statistical  masses  by  the  use  of  frequency 
tables  wherever  possible  and  are  opposed  to  the  use  of 
means.  However,  against  this  tendency  we  can  raise  the 
objection  that  means  are  indispensable  for  certain  pur- 
poses, especially  in  cases  where  we  cannot  work  with  entire 
series,  and  this  is  the  reason  for  their  frequent  use.  It  is 
true,  however,  that  averages  must  always  be  used  with  great 
caution.  The  question  is  not  that  of  eliminating  aver- 
ages but  of  establishing  the  conditions  and  precautions  con- 
cerning their  use. 

To  these  cautions  belongs  the  investigation  of  the  dis- 
persion of  the  series  around  the  mean  used  in  an  individual 
case.  The  value  of  the  mean  depends  essentially  on  the 
dispersion  of  the  items  around  it  and  on  its  position  in 
the  series.  The  question  whether  or  not  the  mean  can  be 
considered   to  be   "  typical,"   can    only   be   answered   by 

251 


252     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

examination  of  the  dispersion  of  the  series  around  it.  If  it 
is  found  that  the  mean  is  "  typical,"  then  its  use  seems  to 
have  a  sufficient  scientific  basis.  In  this  case  the  kind  of 
dispersion  of  the  series  can  most  easily  and,  indeed,  most 
satisfactorily  be  characterized  by  a  single  term  such  as  the 
mean  or  probable  deviation.  If  no  real  typical  mean  with 
symmetric  distribution  of  the  items  around  it  is  present, 
we  may  yet  discover  a  definite  mathematical  law  of  dis- 
tribution, such  as  the  skew  curve  of  error,  to  which  the 
series  corresponds.  Thus,  the  examination  of  the  dispersion 
of  the  items  enables  us  to  determine  exactly  the  theoretical 
value  of  the  mean  and  its  adaptation  for  further  purposes — 
for  instance,  for  purposes  of  comparison.  It  furnishes  us, 
moreover,  with  a  welcome  supplement  to  the  information 
given  by  the  mean  itself.  It  must,  however,  be  remarked 
that  the  non-mathematical  and  the  mathematical  statisti- 
cians in  general  approach  the  examination  of  the  dispersion 
of  the  items  around  their  mean  with  essentially  different 
feelings.  To  the  non-mathematical  statistician  the  series 
itself  is  always  the  most  important  thing.  Of  necessity 
he  works  with  a  mean  in  an  individual  case,  but  he  treats 
it  with  a  certain  distrust  and  in  different  ways  he  tries  to 
obtain  as  many  data  as  possible  about  the  series  itself  and 
uses  these  as  supplementary  to  the  means.  By  establishing 
extreme  cases,  sometimes  also  by  quoting  certain  classes 
in  addition  to  the  mean,  he  tries  to  get  on  safer  ground 
than  the  mean  alone  seems  to  offer.  The  mathematical 
statistician  proceeds  differently.  To  him  the  details  of  the 
series  are  raw  material,  the  quintessence  of  which  is  to  be 
expressed  by  a  typical  mean.  If  he  succeeds  in  establish- 
ing such  a  mean  or  in  discovering  some  law  of  distribution 
to  which  the  dispersion  of  the  items  around  the  mean  cor- 
responds, then  the  mean  in  combination  with  the  law  of 
dispersion  possesses  independent  scientific  meaning  and  is 
more  valuable  than  the  most  accurate  reproduction  of  the 
whole  series. 


PURPOSE  AND  MEANING  OF  DISPERSION       253 

On  the  other  hand,  the  measurement  of  the  dispersion  of 
a  series  can  be  made  not  for  the  purpose  of  supplementing 
the  mean,  but  for  the  purpose  of  estimating  the  value  of 
the  series  itself.  In  this  case  the  measurement  of  the  dis- 
persion is  a  purpose  in  itself  and  the  mean  is  only  com- 
puted to  obtain  a  suitable  basis  for  the  analysis  of  the 
formation  of  the  series.  As  has  already  been  mentioned,^ 
it  is  often  quite  essential  to  start  from  the  mean  in 
obtaining  a  picture  of  the  dispersion  of  the  items.  The 
dispersion  of  a  series  of  quantitative  individual  observa- 
tions indicates  the  kind  and  the  degree  of  variability  of 
an  element  of  measurement  (for  instance,  height,  wage, 
etc.).  The  measurement  of  the  dispersion  of  such  series  is 
of  prime  importance  in  the  field  of  biology,  where  it  leads 
to  the  statistical  comprehension  of  the  laws  of  variation. 
The  dispersion  of  time  series  gives  a  measure  of  the  steadi- 
ness or  the  variability  of  the  phenomena  in  question  during 
the  course  of  time.  The  dispersion  of  geographical  series 
enables  us  to  ascertain  to  what  variations  the  phenomena  in 
question  have  been  subject  in  the  districts  under  con- 
sideration. 

Important  politico-economic  interests  are  often  closely 
connected  with  the  degree  of  variability  of  certain  phenom- 
ena and  numerous  measures  of  public  administration  must 
take  this  into  consideration.  Violent  time  variations  in 
production  or  sale  evidently  must  produce  a  shock  to  the 
economic  organism  which  affects  many  people.  This  is  espe- 
cially true  of  wage  fluctuations.  For  this  reason  the  sliding 
wage  scales,  formerly  in  use  in  various  industries  in  Eng- 
land and  the  United  States,  which  made  the  wages  depend- 
ent on  the  price  fluctuations  of  the  product,  were  usually 
successfully  opposed  by  the  workmen  who  suffered  under 
the  great  wage  fluctuations.  Quetelet  in  his  Physik  dcr 
Gesellschaft     (Physics    of    Society)-     demanded    special 

»  Cf.  p.  125  f. 

*  German  edition  by  Dr.  V.  A.  Riecke  (1838),  p.  C12. 


254     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

measures  to  decrease  the  fluctuations  of  the  price 
of  cereals,  reasoning  as  follows:  "  Since  the  price  of 
cereals  is  one  of  those  causes  which  has  the  greatest  in- 
fluence on  the  mortality  and  reproduction  of  man,  and  since 
this  price  even  to-day  shows  the  greatest  fluctuations,  there- 
fore it  is  the  duty  of  every  far-seeing  government  to 
counteract  all  causes  which  bring  about  those  considerable 
fluctuations  in  the  price  of  cereals  and  consequently  in  the 
'  elements  of  the  social  body. '  ' '  Quetelet  in  the  work  just 
quoted  (p.  613)  finally  came  to  the  following  general  re- 
sult: "  One  of  the  principal  effects  of  civilization  is  the 
constant  narrowing  of  the  limits  within  which  fluctuate 
the  various  elements  upon  which  man  is  dependent.  The 
more  enlightenment  spreads,  the  smaller  become  the  devia- 
tions from  the  mean,  and  the  nearer  we  approach  to  the 
beautiful  and  good."  This  general  conclusion,  however, 
which  is  closely  connected  with  Quetelet 's  conception  of 
the  average  man  as  the  type  of  the  beautiful  and  good, 
does  not  appear  to  be  correct.  In  many  fields  we  are  not 
striving  primarily  for  the  removal  of  the  deviations,  but 
for  progress  all  along  the  line. 

The  degree  of  constancy  must  be  taken  into  consideration, 
especially  with  preliminary  estimates  for  the  future,  which 
individuals  as  well  as  public  administrations  are  often 
forced  to  make,  for  instance,  in  State  or  national  budgets. 
If  a  phenomenon  (such  as  the  price  of  a  certain  commodity) 
has  been  subject  only  to  small  fluctuations  in  the  past,  then 
— under  the  supposition  that  no  new  disturbing  factor  ap- 
pears— a  conclusion  as  to  the  future  is  evidently  more 
surely  drawn  than  with  phenomena  which  have  already 
shown  great  fluctuations  in  the  past. 

The  customs  duties  offer  another  illustration  of  the  im- 
portance of  the  dispersion  of  certain  phenomena  in  govern- 
mental administration.  Many  states  have,  as  is  known, 
largely  supplanted  ad  valorem  duties  by  specific  duties. 
The  ad  valorem  duties  have  been  retained,  however,  in 


PURPOSE  AND  MEANING  OF  DISPERSION       255 

some  states  for  those  goods  whose  value  fluctuates  widely. 

Finally,  it  may  be  mentioned  here  that  statistical  in- 
vestigations concerning  the  degree  of  constancy  of  so-called 
**  moral-statistical  "  phenomena  have  led  to  philosophical 
inferences  of  far-reaching  importance  and  to  violent  con- 
troversies, especially  over  the  question  as  to  whether  social 
phenomena  are  ruled  by  inviolable  laws,  and  the  question 
of  individual  free-will. 

The  investigation  of  the  dispersion  of  statistical  series 
is,  at  any  rate,  necessary,  inasmuch  as  these  series  show  the 
greatest  variety  in  this  respect  and  as  there  are  hardly  two 
series  of  items  of  like  distribution.  Even  the  same  phe- 
nomena, if  presented  statistically  from  different  points 
of  view,  often  result  in  entirely  different  dispersions.  Thus, 
a  phenomenon  may  show  only  small  deviations  from  the 
average  with  regard  to  time,  i.  e.,  it  may  be  constant,  while 
it  shows  great  differences  for  geographical  or  social  differ- 
ences. Again,  other  phenomena  fluctuate  considerably  with 
time,  but  their  structure  remains  the  same  from  year 
to  year. 

Mathematical  and  non-mathematical  statisticians  have 
taken  up  the  investigation  of  the  dispersion  of  statistical 
series  around  their  means.  Mathematical  statisticians  usu- 
ally examine  the  dispersion  of  the  series  with  reference  to 
the  theory  of  errors  of  observation  or  the  theory  of  proba- 
bility. Numerous  works  dealing  with  these  questions  have 
been  written,  so  that  this  subject  forms  one  of  the  most 
highly  developed  chapters  of  mathematical  statistics. 

The  following  discussion  of  the  various  methodological 
questions  of  importance  in  the  measurement  of  the  disper- 
sion will  be  based  on  the  division  of  series  into  the  three 
groups  defined  in  the  first  part  of  this  book. 


CHAPTER  II 

THE    DISPERSION    OF    SERIES    OF    QUANTITATIVE 
INDIVIDUAL    OBSERVATIONS 

A.  MEASUREMENT  AND  PRESENTATION  OF  THE 
DISPERSION  OF  SERIES  OF  QUANTITATIVE  INDI- 
VIDUAL OBSERVATIONS  BY  MEANS  OF  ELEMEN- 
TARY MATHEMATICAL  METHODS 

The  first  of  the  three  groups  of  series  in  our  classification 
contains  the  series  consisting  of  quantitative  individual 
observations,  such  as  measurements  of  age,  length  of  life, 
wages,  etc.  The  simplest  way  of  obtaining  information 
about  the  dispersion  of  such  a  series  around  the  mean  is 
to  ascertain  the  extreme  cases  occurring  in  the  series,  i.  e., 
the  maximum  and  the  minimum  of  the  series  as  well  as 
the  average.  These  values,  therefore,  are  very  frequently 
stated  in  statistical  publications  for  the  purpose  of  con- 
cisely characterizing  the  dispersion  of  wage,  price,  and 
other  series.  The  highest  and  lowest  wages  and  prices  that 
could  be  ascertained  are  stated  as  well  as  the  average  wages 
and  prices.  The  highest  and  the  lowest  temperatures  regis- 
tered during  the  year,  month,  or  day  are  given  in  addition 
to  the  mean  temperature  of  a  locality.  Instead  of  stating 
maximum  and  minimum,  or  perhaps  in  addition  to  this 
statement,  the  distance  between  the  two  extreme  items,  or 
the  distance  of  these  items  from  the  average,  may  also 
be  computed  as  a  supplement  to  the  latter.  Furthermore, 
the  distance  of  the  extreme  items  from  each  other  or  from 
the  average  is  sometimes  expressed  as  a  percent  of  the 
average  or  of  the  highest  or  lowest  item.^ 

*  We  must  distinguish  the  method  of  characterizing  a  series  given 
in  its  totality  by   the   statement  of  the  average  and  the  extreme 

256 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      257 

The  extremes  of  a  series  possess  significance  in  that  they 
give  the  ''  range  "  within  which  all  observations  of  the 
series  fall.*  But  knowledge  of  the  "  range  "  gives  us  no 
information  of  the  distribution  of  the  items  within  its 
limits.  This  distribution  may  vary  widely  even  though 
maxima  and  minima  are  the  same.  On  the  other  hand, 
series  with  different  extremes  may  have  practically  the 
same  conformation  or  dispersion.  It  is  to  be  noted  that 
the  sizes  of  maxima  and  minima  depend  largely  upon  the 
number  of  observations.  The  greater  the  number  of  obser- 
vations, the  greater  is  the  possibility  of  obtaining  greater 
deviations  from  the  average.  Thus,  the  limits  within  which 
the  heights  of  the  inhabitants  of  a  village  fluctuate  are  not 
the  standard  limits  for  the  whole  country.     Among  the 

values,  from  those  cases  where  only  the  average  and  the  maximum 
and  the  minimum  (or  one  of  these  extreme  values)  are  ascertained 
in  the  observation.  Such  cases  are  not  rare.  Thus,  in  the  Austrian 
Labor  Bureau's  investigation  of  the  condition  of  workmen  in  the 
Ostrau-Karwin  coal  district  not  only  the  detailed  wages  of  the 
miners,  but  also,  for  purposes  of  comparison,  wage  data  in  small 
industrial  establishments  and  the  wages  of  agricultural  and  forest 
workmen  were  ascertained.  That  is,  the  amount  of  the  highest,  the 
lowest,  and  the  normal,  or  average,  cash  wages  of  the  workmen  in 
question  were  asked  for  and  classified  by  industry  or  nature  of  land. 
(Cf.  Arbeiterverhaltnisse  im  Ostrau-Karwiner  Steinkohlenreviere,  Pt. 
I,  p.  xvii.)  In  the  same  investigation  the  usual  retail  prices  of 
certain  commodities  were  also  ascertained  and,  in  case  of  great  price 
fluctuations  during  the  year,  their  lowest  and  highest  prices. 

*  The  extreme  cases  are  sometimes  defined  more  closely  on  account 
of  their  significance.  If  a  phenomenon  which  changes  in  the  course 
of  time  is  characterized  by  the  statement  of  its  average  and  its 
maximum  and  minimum,  then  frequently  the  date  is  also  given  on 
which  maximum  and  minimum  have  occurred  (this  is  frequently 
done  in  the  Tabellen  zur  Wiihrungsstatistik,  published  by  the 
Austrian  Treasury  Department).  Often  only  one  of  the  extreme 
cases  is  of  significance.  Thus,  if  factories,  sick  reliefs,  etc.,  are 
arranged  and  presented  according  to  size,  the  largest  sick  relief  or 
the  largest  establishment,  etc.,  are  sometimes  characterized  by  the 
statement  of  particular  criteria   (name,  location,  etc.). 


258     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

larger  population  of  the  whole  country  deviations  from 
the  mean  are  probable  which  are  entirely  outside  of  the 
range  for  a  single  village.  In  spite  of  this,  the  distribution 
of  the  items  about  the  mean  in  both  cases  may  be  essentially 
the  same.^ 

Certain  information  about  the  dispersion  in  statistical 
series  of  quantitative  individual  observations  may  also  be 
obtained  by  computing  several  averages  instead  of  one. 
From  the  relative  position  of  the  various  averages  im- 
portant conclusions  about  the  conformation  of  the  series 
may  be  drawn.  If  arithmetic  mean,  mode,  and  median 
coincide,  or  differ  but  slightly,  then  it  is  certain  that  the 
series  is  symmetrical.  If  they  differ  materially,  it  is  of 
special  importance  to  know  whether  the  mode  lies  above 
or  below  the  arithmetic  mean  or  the  median,  and  how  much 
it  differs  from  them.  Concerning  the  relation  between  the 
median  and  the  arithmetic  mean,  the  fact  that  the  arith- 
metic mean  lies  below  the  median  indicates  that  the  devia- 
tions below  the  median  are  greater  than  those  above;  vice 
versa,  if  the  arithmetic  mean  lies  above  the  median,  the 
deviations  above  the  median  are  evidently  larger. 

The  median — found  by  bisecting  the  series — may  be 
supplemented  by  stating  those  values  which  result  from 
dividing  the  series  into  more  than  two  groups  of  equal 
frequency.  Thus,  the  quartiles  originate  from  a  division 
of  the  series  into  four  groups  of  equal  size.  Stating  the 
first  and  third  quartiles  (the  second  quartile  being  the 
median)  of  a  series  enables  us  to  see  within  what  limits  half 
the  items  are  located.®    Furthermore,  the  deciles  (or  some 

"  Fechner,  especially,  has  taken  up  the  work  on  the  connection 
between  the  number  of  observations  and  the  sizes  of  the  extreme 
deviations  from  the  mean,  and  has  developed  laws  of  this  relation 
for  series  which  correspond  to  the  normal  or  to  the  unsymmetrical 
Gaussian  law.  (KoUektivmasslehre,  Chap.  XX,  "The  Laws  of  the 
Extremes,"  pp.  321-338.) 

'  Thus,  in  the  American  special  report,  Employees  and  Wages 
(1903),  in  Chap.  II,  "Analysis  of  Occupational  Comparison"    (pp. 


I 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      259 

of  them),  found  by  dividing  the  series  into  ten  equal 
groups,  may  be  given.  If,  in  a  series,  the  mode  of  which 
is  stated,  other  "  secondary  "  points  of  concentration  can 
be  foimd,  the  statement  of  these  is  of  special  significance.'' 

A  third  way  to  give  more  information  of  a  series  which 
cannot  be  presented  in  detail  than  is  possible  by  merely 
stating  one  mean,  is  to  present  certain  characteristic  classes 
— be  they  connected  with  the  mean  or  independent  of  it — 
as  weU  as  the  mean. 

If,  besides  the  median,  other  values  originating  from 
the  division  of  the  series  into  more  than  two  equal  groups 
be  taken,  such  as  the  quartiles  or  the  deciles,  then  they 
describe  the  classes  adjoining  the  middle  of  the  series, 
for  the  quartiles  indicate  between  what  limits  below  and 
above  the  median  one-quarter  of  the  items  are  located, 
the  deciles  indicate  between  what  limits  one,  two,  three, 
or  four-tenths  of  the  items  are  located  above  or  below 
the  median.  Likewise,  if  we  desire  to  supplement  an 
arithmetic  mean  or  a  mode,  we  may  compute  and  state 
within  what  limits  are  located  one-quarter  or  one,  two,  three, 
or  four-tenths,  or  any  other  fraction  of  the  items,^     The 

xxix-xcix),  the  comparison  of  the  wages  of  the  workmen  of  various 
occupations  is  made  for  the  years  1890  and  1900  by  comparing  the 
median  and  the  two  quartiles  of  each  of  the  wage  series  referring 
to  the  two  years  mentioned.     (Cf.  ibid.  p.  xxviii.) 

*  Fechner  (Kollektivmasslehre,  Chap.  XIX,  "The  Laws  of  Asym- 
metry," pp.  294-306)  has  developed  a  number  of  laws  for  the  relation 
existing  between  the  various  means  in  series  which  correspond  to 
the  unsymmetrical  Gaussian  law.  He  has  determined  theoretically 
the  intervals  between  these  values,  their  relative  positions,  and  the 
peculiarities  of  the  deviations  from  the  arithmetic  mean  and  from, 
the  mode.  Thus,  he  has  determined  a  law  of  positions,  according 
to  which,  if  the  unsymmetrical  Gaussian  law  holds,  the  arithmetic 
mean  and  the  median  always  lie  on  the  same  side  of  the  mode  in  such 
a  manner  that  the  median  falls  between  the  arithmetic  mean  and 
the  mode.     (Cf.  also  Fechner,  "  Ausgangswert,  etc.,"  p.  11.) 

*  Thus,  March  in  "  Quelques  examples  de  distribution  des  salaires  " 
(Journal  de  la  Societe  de  Statistique  de  Paris,   1898,  p.  201)    hag 


260     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

wider  the  limits  between  which  a  given  fraction  of  items 
is  located,  the  greater  evidently  is  the  dispersion.  On  the 
other  hand,  we  may  ascertain  how  many  items  are  within 
definite  limits  above  and  below  the  average — for  instance, 
how  many  items  do  not  deviate  more  than  10^  from  the 
average.  The  greater  the  number  of  these  items  relatively, 
the  closer  the  series  will  be  condensed  about  the  mean  and 
the  smaller  its  dispersion. 

Aside  from  classes  joining  the  average,  classes  indepen- 
dent of  the  average  naturally  can  also  be  presented  to 
supplement  it.  It  may  be  especially  significant  to  state 
the  extreme  classes — for  instance,  the  extreme  wage  classes 
of  a  definite  width  together  with  the  average  wage.  An 
even  more  concise  characterization  is  obtained  by  merely 
stating  the  averages  of  the  extreme  classes.  This  is  often 
better  than  the  statement  of  the  extreme  cases  which  may 
be  far  removed  from  all  the  other  items.^  But  then  it 
must  be  mentioned,  if  possible,  upon  how  many  items 
the  averages  are  based,  since  this  cannot  be  deduced  from 
the  mere  statement  of  the  averages.^"  The  most  expedient 
way  of  supplementing  the  average  can  evidently  be  decided 
only  from  the  peculiarities  of  the  given  case.     The  purpose 

computed  the  intervals  from  the  mode  within  which  30;?,  50^,  etc., 
of  the  wages  which  he  is  discussing  are  located. 

°  Therefore,  the  computation  of  the  average  of  the  maximum  and  the 
minimum,  which  we  sometimes  meet  with,  has  only  little  value. 

'°  Cf.  Bowley,  Elements  of  Statistics,  2nd  ed.,  p.  93  ff.     Bowley 

has  also  proposed  to  use  the  formula.  ^""^^  (in  which  Q,  and  Qa 

Qs-j-Qi 
denote  the  two  quartiles)   for  measuring  the  dispersion  of  series  of 

quantitative  individual  observations,  especially  for  purposes  of  com- 
parison. This  fraction  increases  with  the  distance  between  the  two 
quartiles  and  clearly  expresses  changes  in  the  series.  Its  value 
always  lies  between  0  and  1.  By  means  of  this  formula  Bowley 
has  computed  the  dispersion  of  the  wages  of  English  mechanics  for 
1862  and  1890  and  found  the  number  0.093  for  the  first  year  and 
0.062  for  the  second  year.  Thus  the  dispersion  of  the  wages  had 
decreased.     (Elements,  p.  136.) 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      261 

of  the  investigation  in  hand  is  the  determining  factor  in 
the  selection  of  any  particular  method. 

Instead  of  gathering  various  details  from  the  series  to 
be  characterized,  in  order  to  supplement  the  average,  we 
may  obtain  a  single  numerical  expression  for  the  dispersion 
of  the  series  by  computing  an  average  of  the  deviations  of 
the  items  from  the  average  of  the  series  {fluctuation  num- 
ber). Recognizing  the  desirability  of  such  a  measure  the 
Statistical  Congress  at  The  Hague  in  1869  passed  the  follow- 
ing resolution:  "  Le  congres  est  d'avis  qu'il  est  a  desirer 
qu'on  calcule  non  seulement  les  moyennes,  mais  aussi 
le  nombre  d 'oscillations,  afin  de  connaitre  la  deviation 
moyenne  des  nombres  d'une  serie  de  la  moyenne  de  cette 
serie  meme."  However,  according  to  G.  von  Mayr,  it  is 
not  sufficient  to  compute  the  average  deviation,  but  the 
latter  must  be  expressed  as  a  percentage  of  the  average 
of  the  series.^^  But  it  does  not  appear  to  be  always  neces- 
sary to  find  this  percentage.  The  absolute  size  of  the 
average  deviation  may  be  significant  and  sufficient  to  sup- 
plement the  average.  Thus,  the  mean  height  of  a  given 
population  may  be  given  and,  as  a  supplement  to  this 
average,  it  may  be  stated  how  many  centimeters  the  in- 
dividuals belonging  to  this  population  deviate  on  the  aver- 
age above  or  below  this  mean.  According  to  6.  von  IMayr 
we  ought  to  state  the  percentage  that  the  average  deviation 
bears  to  the  average  height. 

The  question,  whether  the  average  deviation  should  be 
stated  as  an  absolute  number  or  as  a  percentage  of  the 
average  of  the  series,  has  been  fully  discussed  by  the  mathe- 
matical statisticians.  They  are  of  different  opinions.  Lexis 
is  decidedly  opposed  to  the  presentation  of  the  probable 
and  average  deviations  computed  for  typical  series  of  meas- 
urements as  a  percent  of  the  average  of  the  series.  In 
his  opinion  there  is  generally  no  reason  for  the  assumption 
that  the  average  (and  also  the  probable)  deviation  depends 

"  Theoretische  Statistik,  p.  100. 


262     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

in  any  way  on  the  absolute  size  of  the  base  value  (the 
average  of  the  series).  "  Suppose  that  the  heights  and  the 
chest  measures  of  a  number  of  ten-year-old  boys,  and  of 
a  number  of  fully  grown  men  have  been  taken.  Presuma- 
bly, the  former  series  will  result  in  a  greater  average  devia- 
tion from  the  mean  than  the  latter,  and  it  is  this  difference 
which  corresponds  to  the  physical  difference  in  the  stability 
of  the  two  anthropometric  values.  If  we  express  the  two 
deviations  as  percentages  of  the  respective  base  values,  then 
the  divergence  of  these  measures  of  dispersion  becomes 
considerably  greater  than  that  of  the  absolute  deviations; 
but  the  former  cannot  be  compared  with  each  other,  while 
the  latter  are  in  an  inverse  proportion  to  the  precision,  and 
thus  may  be  considered  to  be  a  direct  and  analogous  expres- 
sion of  the  dispersion. ' '  ^^ 

Fechner  takes  an  essentially  different  stand.  He  dis- 
tinguishes between  repeated  measurements  of  the  same 
object  affected  with  accidental  errors  of  observation  (physi- 
cal and  astronomic  measurements)  and  measurements  of 
various  similar  objects  in  a  collective  group.  If  series  of 
the  first  kind  are  given,  then  the  size  of  the  deviation  of 
the  items  from  the  mean  is  independent  of  the  size  of  the 
mean  (i.  e.,  the  object  measured)  and  the  absolute  size 
of  the  average  deviation  must  be  stated.  However,  in  series 
of  the  second  kind  it  is  Fechner 's  opinion  that  the  devia- 
tions usually  depend  on  the  average  size  of  the  collective 
object  in  question,^^  and  from  this  he  draws  the  conclusion 

'^  Abhandlungen,  etc.,  VIII,  "On  the  Theory  of  the  Stability  of 
Statistical  Series,"  p.  173  f. 

""Generally  speaking,  errors  of  observation  are  essentially  inde- 
pendent of  the  size  of  the  object  measured,  at  least  with  measure- 
ments of  space  and  on  the  assumption  that  the  measuring  instru- 
ments are  not  changed.  The  errors  of  observation  are  larger  when 
we  measure  a  mile  than  when  we  measure  a  foot,  it  is  true,  but 
only  because  more  and  more  complicated  operations  are  necessary 
to  measure  the  former,  but  the  errors  of  observation  when  measuring 
a  high  thermometric  or  barometric  registration  are  generally  not 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      263 

that  with  collective  objects  the  average  deviation  must  be 
expressed  as  a  percent  of  the  mean  of  the  series,  in  order 
to  eliminate  the  influence  of  its  size. 

Concerning  the  question,  from  what  series  may  we  com- 
pute measures  of  fluctuations,  von  Mayr  (loc.  cit.)  says: 
"  The  measure  of  fluctuation  can  be  computed  arithmetic- 
ally for  any  numerical  series.  It  has  a  statistical  value, 
however,  only  in  certain  series,  particularly  those  in  which 
the  items  have  the  character  of  similarity,  especially  in 
the  sense  that  they  present  the  process  of  social  phe- 
nomena developing  in  equal  time  periods."  Evidently  von 
Mayr  has  time  series  primarily  in  mind.    But  even  series 

larger  than  when  measuring  a  low  one.  The  variations  of  collective 
objects,  however,  are  essentially  dependent  on  their  sizes,  if  this  be 
understood  along  the  lines  of  the  following  illustrations:  A  flea 
is  a  small  creature,  and  therefore  the  deviations  of  the  individual 
fleas  from  the  average  flea  are  small  on  the  average,  being  only 
fractions  of  the  mean  size,  and  the  total  diS'erence  between  the 
largest  and  the  smallest  flea  is  but  slight.  The  mouse  is  much 
larger  than  the  flea,  the  horse  again  much  larger  than  the  mouse, 
a  tree  much  larger  than  an  herb,  etc.,  and  everywhere  a  correspond- 
ing observation  is  made,  i.  e.,  that  the  deviations  of  the  individual 
mice  from  the  average  mouse  are  greater  on  the  average  than  those 
of  the  individual  fleas  from  the  average  flea,  etc.  This  dependence 
of  the  average  sizes  of  the  deviations  on  the  average  sizes  of  the 
objects  can  also  be  explained  by  the  fact  that  the  interior  and 
exterior  modifying  causes  find  more  points  of  attack  on  large  than 
on  small  objects.  The  quality  of  the  object  is  also  of  significance 
because  of  the  greater  or  smaller  facility  with  which  it  yields  to 
the  modifying  influence;  and  the  accessibility  to  exterior  modifying 
influences  may  diflfer  with  circumstances.  Therefore,  an  exact  pro- 
portionate relation  of  the  average  size  of  the  deviations  to  the  aver- 
age size  of  the  objects  cannot  be  expected  a  priori.  But  in  any  case 
the  sizes  of  the  objects  are  principal  factors  influencing  the  sizes 
of  their  deviations."  (Kollektivmasslehre,  p.  77  f.)  Also  of.  Fech- 
ner,  "  Ausgangswert,  etc.,  pp.  14  and  16.  Georg  Duncker  diff"ers 
from  Fechner  in  regard  to  the  dependence  of  the  size  of  the  devia- 
tions on  the  size  of  the  objects  measured.  He  denies  the  presence 
of  such  a  dependence  on  the  basis  of  his  biological  investigationa. 
(Cf.  Die  Methode  der  Variationsstatistik,  p.  40.) 


264     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

of  quantitative  individual  observations  are  not  excluded 
from  the  measurement  of  dispersion  by  means  of  an  average 
deviation.  It  is  for  just  such  series  that  mathematical 
statisticians  have  evolved  their  methods  of  measuring  dis- 
persion by  the  computation  of  various  kinds  of  average 
deviations.^* 

The  measure  of  fluctuation  usually  taken  is  the  arith- 
metic mean  of  the  deviations  from  the  arithmetic  mean  of 
the  series.  However,  other  means  may  also  be  computed 
from  the  deviations  of  the  items  from  the  arithmetic  mean 
of  the  series.  Furthermore,  the  deviations  of  the  items 
are  not  necessarily  measured  from  the  arithmetic  mean 
of  the  series;  on  the  contrary,  some  other  mean  of  the 
series  may  be  made  the  starting  point.  The  choice  of  the 
mean  of  the  series,  from  which  the  deviations  are  measured, 
and  the  choice  of  the  measure  of  fluctuation  which  is  com- 
puted from  the  deviations,  are  theoretically  interdependent. 
The  arithmetic  mean  of  a  series  of  items  is  characterized 
by  the  fact  that  the  sum  of  the  squares  of  the  deviations 
from  it  is  a  minimum;  the  median,  on  the  other  hand,  is 
characterized  by  the  fact  that  the  sum  of  the  simple  devia- 
tions of  the  items  from  the  median  is  a  minimum,  i.  e., 

^*  V.  Mayr  ( Tlieoretische  Statistik,  p.  100)  remarks  that  a  knowl- 
edge of  the  deviations  possesses  the  least  objective  interest  for  the 
"  most  pronounced  form  of  typical  series."  By  the  "  most  pro- 
nounced form  of  typical  series"  v.  Mayr  (ibid.  p.  90)  means  those 
series  in  which  the  arranged  items  lie  symmetrically  above  and 
below  the  mean;  the  items  are  considered,  so  to  speak,  to  be  inac- 
curate reproductions  of  a  constant  base  value,  which  in  the  actually 
observed  phenomena  is  expressed  with  merely  accidental  deviations. 
However,  it  is  not  quite  clear  why  the  knowledge  of  the  deviations 
of  the  items  of  such  pronounced  typical  series  should  offer  the 
smallest  amount  of  objective  interest,  since  even  with  symmetrical 
distribution  of  the  items  about  the  mean  and  with  a  central  mode 
of  the  curve  the  sizes  of  the  deviations  may  vary  extremely.  It 
is  in  the  case  of  a  symmetrical  distribution  of  the  items  that  the 
indices  of  fluctuation,  as  is  explained  later,  are  of  the  greatest 
scientific  value. 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      265 

smaller  than  the  sum  of  the  deviations  of  the  items  from 
any  other  value.  Therefore,  it  would  really  be  theoretically 
correct  to  base  the  average  deviation  (the  arithmetic  mean 
of  the  simple  deviations)  on  the  median  of  the  series;  if  the 
arithmetic  mean  of  a  series  is  to  be  supplemented,  then  the 
mean  square  of  the  deviations  ought  to  be  used,  which  is 
found  by  dividing  the  sum  of  the  squares  of  the  deviations 
of  the  items  by  their  number  and  then  finding  the  square 
root  of  the  quotient.^^  As  a  matter  of  fact,  the  mean 
square,  as  defined  above,  is  generally,  although  not  ex- 
clusively, used  in  the  theory  of  error  as  a  measure  of  the 
dispersion  about  the  arithmetic  mean  of  the  series;  it  is 
called  the  standard  deviation.  Laplace  has  used  the 
mean  error  of  the  simple  deviations,  although  he 
measured  these  deviations  in  the  usual  way  from  the 
arithmetic  mean  of  the  series.^*'  In  elementary  mathemat- 
ical statistics  the  mean  square  is  not  known  at  all,  and 
certainly  never  will  be  widely  used  for  averaging  the  devia- 
tions from  the  arithmetic  mean.  Likewise  the  median  of 
the  deviations,  i.  e.,  that  deviation  which  lies  in  the  center 
of  the  series  of  deviations  arranged  according  to  size,  the 
"  probable  "  deviation  which  plays  an  important  role  in 
mathematical  statistics,  is  rarely  used  in  elementary  mathe- 
matical statistics. 

In  order  to  be  able  to  judge  the  methodological  value 
of  measures  of  dispersion  we  must  keep  in  mind  that  while 
they  offer,  as  averages  of  the  deviations  of  the  items  from 
the  mean  of  the  series,  a  comprehensive  numerical  expres- 
sion for  these  deviations,  they  are  averages,  and  hence  can 
never  give  a  complete  picture  of  these  deviations  and  their 
sizes.  This  is  a  general  truth  whatever  be  the  average  of 
the  series  from  which  the  deviations  are  measured,  and 
whatever  be  the  average  which  is  computed  from  the  devia- 
tions themselves, 

"  Cf,  above,  Pt.  II,  note  4, 

*'  Cf.  Feehner,  "  Ausga^g3^Yert,  etc.,"  p.  54, 


266     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

It  is  a  peculiar  quality  of  every  measure  of  dispersion 
that  in  its  computation  reaUy  two  groups  of  deviations 
are  united  and  characterized  by  means  of  a  single  average. 
For  there  are  the  deviations  of  the  items  above  the  average 
of  the  series  and  the  deviations  of  the  items  below  this 
average.  The  numbers  of  the  items  above  and  below  the 
average  usually  differ  if  we  start  from  the  arithmetic  mean 
or  the  mode ;  if  we  start  from  the  median,  then,  according 
to  the  definition  of  this  average,  the  numbers  of  the  positive 
and  negative  deviations  are  equal;  but  the  sizes  of  the 
deviations  above  and  below  the  median  may  vary.  There- 
fore, with  good  reason,  we  could  consider  the  positive 
deviations  and  the  negative  deviations  of  the  items  from 
the  average  of  the  series  to  be  two  separate  series  and 
compute  the  mean  deviations  separately  for  each  set. 
However,  this  is  not  usually  done,  but  the  deviations  of 
all  the  items  are  combined  or  united,  so  to  speak,  into  a 
series,  and  this  series  is  characterized  by  a  single  mean, 
the  average  deviation.  If  the  positive  deviations  are  equal 
to  the  negative  deviations,  then  there  is  no  objection  to 
their  combination  and  presentation  by  means  of  a  single 
average  (the  mean  deviation).  But  if  these  deviations  are 
not  equal,  then,  by  combining  them  and  computing  a  com- 
mon mean,  a  value  is  obtained  which  correctly  characterizes 
neither  the  deviations  above  nor  those  below  the  average 
of  the  series.  Therefore  the  question  is,  when  are  the  devia- 
tions of  the  items  above  the  average  of  a  statistical  series 
equal  to  the  deviations  of  the  items  below  the  average? 
Such  a  coincidence  occurs  when  the  whole  statistical  series 
is  distributed  symmetrically  about  the  average  from  which 
the  deviations  of  the  items  are  measured.^^  Therefore,  only 
in  this  case  is  the  computation  of  a  numerical  measure  of 
dispersion,  combining  all  the  deviations  from  the  mean 
into  a  single  average,  entirely  free  from  objection.     This 

*^  It  is  not  necessary  that  the  series  contain  a  central  mode  as 
the  theory  of  error  presupposes,  but  it  is  usually  the  case. 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      267 

measure  of  dispersion  naturally  varies  according  to  the 
distribution  of  the  items  on  both  sides  of  the  mean.  If 
the  items  of  the  series  are  not  distributed  symmetrically 
about  the  mean  of  the  series  from  which  the  deviations 
are  measured,  then  the  measure  of  dispersion  computed 
for  the  series  has  only  little  value  and  may  actually  lead 
to  error.  Let  us  imagine  a  series  which  has  a  mode  on 
one  side  of  the  arithmetic  mean — but  not  very  far  from 
it — while  on  the  other  side  of  the  arithmetic  mean  fewer, 
but  very  marked,  deviations  are  present,  so  that  the 
conformation  of  the  series  above  and  below  the  arithmetic 
mean  is  not  at  all  the  same.  If,  starting  from  the  arith- 
metic mean,  we  compute  a  measure  of  dispersion  for  this 
series,  we  obtain  a  small  number  and,  merely  knowing  this 
number,  we  are  likely  to  assume  that  the  whole  series  is 
condensed  about  the  arithmetic  mean  with  little  disper- 
sion.^'^^ 

Therefore,  with  series  of  irregular  conformation  it  is  bet- 
ter not  to  take  an  average  of  all  the  deviations  as  a  measure 
of  dispersion.  But,  under  certain  circumstances,  it  may  be 
both  allowable  and  expedient  to  compute  separate  measures 
of  dispersion  for  the  items  above  and  below  the  averages  of 
such  series.  This  is  advisable  if  the  structure  of  the  series 
differs  on  the  two  sides  of  the  base  value  while  each  side, 
taken  alone,  shows  a  regular  conformation.  Such  a  dis- 
persion is  found  quite  frequently  if  the  mode  of  a  series 

*'aThe  author  is  not  in  accord  with  the  general  tendency  in  lay- 
ing so  much  stress  on  symmetry  as  a  prerequisite  for  the  use 
of  measures  of  dispersion.  Too  great  insistence  that  statistical 
data  must  conform  to  mathematical  rules  is  fatal;  it  destroys  the 
usefulness  of  the  science  of  statistics.  Even  though  series  are  not 
symmetrical,  measures  of  dispersion  may  be  profitably  employed  to 
characterize  them.  Of  course,  in  computing  the  arithmetic  average 
deviation  from  the  arithmetic  average  of  a  series  all  deviations  are 
considered  positive;  in  computing  the  standard  deviation  the  squares 
of  the  deviations  only  contribute  to  the  size  of  the  resulting  measure 
of  dispersion. — Translatob. 


268     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

of  individual  observations  is  made  the  base  value.  In  this 
ease  we  may  try  to  express  the  deviations  of  the  parts 
of  the  series  on  both  sides  of  the  modes  by  separate 
averages  of  deviations.  If  not  even  the  conditions  for 
this  kind  of  expression  be  given,  then  we  must  try  to  char- 
acterize the  dispersion  of  the  series  as  well  as  possible  in 
some  other  way:  by  stating  the  extremes,  by  computing 
various  averages,  by  presenting  certain  classes,  by  stating 
how  many  items  lie  above  and  how  many  below  the  arith- 
metic mean  or  the  mode,  etc.  Finally,  it  must  be  noted 
that  there  are  series  of  items  which  are  distributed  about 
the  mean  with  no  regularity,  but  which  have  some  other 
regular  characteristic  structure.  The  nature  of  such  a 
series  cannot  be  ascertained  by  investigating  the  dispersion 
of  the  series  about  its  mean,  but  the  peculiar  regular 
conformation  must  be  ascertained  and  expressed  by  other 
methods  which  are  discussed  in  Appendix  I. 

It  must  be  mentioned  in  this  connection  that  the  kind  of 
dispersion  of  series  of  quantitative  individual  observations 
essentially  depends  on  whether  or  not  the  series  is  homo- 
geneous. Series  of  a  heterogeneous  make-up  usually  show 
an  irregular  conformation.  They  generally  contain  several 
points  of  concentration  of  equal  or  varying  importance — 
which  correspond  to  the  modes  of  the  combined  con- 
stituents— and  there  is  no  symmetric  distribution  about 
any  mean.  If  such  series  are  decomposed  into  the 
more  homogeneous  constituents  of  which  they  consist,  then 
constituent  series  result,  each  of  which  frequently  shows 
only  one  mode,  centrally  located,  which  approximately 
coincides  with  the  arithmetic  mean  and  the  median;  the 
items  being  distributed  with  a  certain  symmetry  and  regu- 
larity about  the  means  of  each  homogeneous  constituent 
series.  If  the  wages  of  all  the  workmen  of  a  given  district 
are  combined  in  a  statistical  series,  a  very  irregular  con- 
formation is  usually  the  result.  Since  sex  generally  has 
a  decisive  influence  on  the  amount  of  wages,  the  total 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      269 

series,  giving  the  wages  of  men  and  women,  probably  con- 
tains two  points  of  concentration  for  the  two  sexes.  Even 
if  men  and  women  are  treated  separately  several  points 
of  concentration  may  occur,  caused  by  combining  workmen 
of  different  occupations.  In  addition,  the  wages  of  the 
skilled  and  unskilled  laborers  of  the  same  occupation  may 
stand  forth  as  separate  groups.^''''  But  if  all  the  elements  of 
differentiation  influencing  larger  groups  of  workmen  are 
taken  account  of,  and  if  essentially  homogeneous  constituent 
series  are  formed  by  suitably  decomposing  the  entire  series 
into  constituent  series  which  contain  only  individual  differ- 
ences, then  the  occurrence  of  several  points  of  concentration 
is  not  probable,  and  we  can  count  on  a  symmetric  distribu- 
tion. 

Besides  attempting  to  secure  greater  homogeneity  of  kind 
we  may  also  strive  for  greater  homogeneity  of  space  and 
time  and,  in  this  way,  obtain  an  improvement  of  the  con- 
formation of  the  series.  Items  originating  in  different 
countries  or  districts  frequently  give  rise  to  varying  aver- 
ages and,  therefore,  in  case  of  their  combination,  to  several 
points  of  concentration  which  may  be  removed  by  suitably 
decomposing  the  series.  Measurements  of  the  stature  of 
individuals  of  various  nationalities  result  in  an  irregular 
series,  while  the  stature  of  the  inhabitants  of  the  same 
country  are  distributed  regularly  about  their  aver- 
age. 

However,  there  may  also  occur  homogeneous  series  of 
unsymmetrical  conformation.  In  biology  and  anthropology 
such  series  generally  indicate  an  evolution,  to  be  considered 
an  improvement  or  a  degeneration  according  to  the  nature 
of  the  case.     The  fact  that  evolution  is  the  cause  of  an 

*'b  Likewise,  Prof.  H.  L.  IMoore  has  shown  that  union  and  non- 
union workmen  belong  in  separate  categories  as  regards  wages  (see 
Laws  of  Wages,  p.  189).  Age  of  employees  and  size  of  establishment 
also  appear  to  give  significant  wage  categories  (Laws  of  Wages,  p. 
143 ) . — Tbanslatob. 


270     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

unsymmetrical  conformation  is  explained  by  Lexis  ^^  in 
the  following  way:  A  certain  percentage,  say,  half  of 
the  totality,  is  still  distributed  regularly  about  the  original 
type,  while  the  rest  shows  another  distribution  on  account 
of  external  influences  or  other  causes.  If  there  is,  for 
example,  degeneration,  caused  by  injurious  influences,  we 
may  assume  that  the  subnormal  groups  are  affected  by 
these  influences  to  an  extent  measured  by  their  distance 
below  the  normal  type  and  that  the  supra-normal  groups 
are  influenced  comparatively  less.  By  combining  the  de- 
generate half  with  the  stable  half  an  unsymmetrical  dis- 
tribution of  the  totality  results,  in  which  groups  below  the 
normal  are  more  extended  than  the  groups  above  the 
normal. 

B.  MEASUREMENT  OF  THE  DISPERSION"  OF  SERIES 
OF  QUANTITATIVE  INDIVIDUAL  OBSERVATIONS 
FROM  THE  STANDPOINT  OF  THE  THEORY  OF  ERROR 

When  judging  the  dispersion  of  series  of  quantitative 
individual  observations  from  the  standpoint  of  the  theory 
of  error  the  point  at  issue  is  to  ascertain  whether  the  dis- 
tribution of  the  items  corresponds  to  the  normal  (sym- 
metrical) law  of  error  as  defined  by  the  Gaussian  law,  or 
to  any  generalized  (unsymmetrical)  law  of  error. 

The  normal  law  of  error  was  established  and  formulated 
originally  on  the  basis  of  repeated  observations  of  the 
same  object,  especially  such  as  repeated  measurements  of 
the  same  object  in  the  astronomical,  physical,  or  geodetic 
fields.  We  know  empirically  that  repeated  measurements 
of  an  object,  the  size  of  which  is  to  be  ascertained,  do  not 
completely  coincide.  The  various  measurements  are  affected 
with  varying  "  accidental  "  errors  of  observation.  How- 
ever, the  series  formed  by  the  individual  measurements 
shows  a  regular  characteristic  conformation.     The  series  is 

'*  Cf.  "  Anthropologie  und  Anthropometrie "  in  the  Handw.  d. 
Staatsw.,  2nd  ed.,  Vol.  I,  p.  397,  and  Abhandlungen,  etc.,  p.  124. 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      271 

distributed  symmetrically  about  the  arithmetic  mean, 
the  most  probable  value  of  the  object  measured,  the  items 
being  crowded  most  densely  around  the  arithmetic  mean 
and  becoming  rarer  the  farther  distant  they  are  from  it. 
The  frequency  of  the  various  measurements  is  a  function 
of  their  distance  from  the  arithmetic  mean  or,  in  other 
words,  the  frequency  of  the  varying  deviations  from  the 
arithmetic  mean  is  a  function  of  the  size  of  these  deviations. 
Gauss  was  the  first  to  investigate  the  probability  of  the 
varying  deviations,  and  has  formulated  the  law  of  dis- 
tribution (called  after  him  the  Gaussian  probability  in- 
tegral) for  the  dispersion  of  the  items  about  the  arithmetic 
mean.^^* 

The  sizes  of  the  deviations  in  a  series  of  repeated  meas- 
urements of  the  same  object  depend  on  the  precision  of 
the  individual  measurements.  The  greater  this  precision, 
the  smaller  are  the  accidental  errors  and  the  more  densely 
are  the  items  crowded  together.  Consequently,  the  graphic 
presentation  of  the  series  results  in  a  curve  which,  with 
great  precision  of  the  items,  extends  only  over  a  small  part 
of  the  axis  of  the  abscissas  and  declines  abruptly  on  both 
sides  of  the  arithmetic  mean,  while  the  curve  which  is 
based  upon  less  precise  measurements  is  more  extended, 
and  the  items  show  greater  deviations  on  both  sides  of  the 
average. 

The  dispersion  of  a  series  of  repeated  measurements  of 
the  same  object,  obeying  the  Gaussian  law,  can  be  character- 
ized by  a  single  expression  obtained  by  computing  an 
average  of  the  deviations  from  the  arithmetic  mean.  As 
such  measures  of  dispersion  the  error  of  mean  square 
(standard  deviation),  the  average  error,  and  the  probable 
error   as  well   as  the  modulus  may   be   used.^^     Certain 

*'a  See  the  equation  of  the  Gaussian  cun^e  on  p.  166,  footnote  37a. 

'"  The  modulus  is  to  be  computed  as  the  square  root  of  twice  the 
quotient  of  the  sum  of  the  deviations  from  the  arithmetic  mean 
divided  by  the  number  of  items.     Edgeworth  has  proposed  the  term 


272     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

mathematical  relations  exist  between  these  measures  of 
dispersion,  so  that  one  can  be  computed  from  the 
other.2o 

A  picture  of  the  dispersion  of  the  items  of  a  symmetrical 
series,  obeying  the  normal  law  of  error,  about  the  arith- 
metic mean  of  the  series  is  given  in  the  following  table 
taken  from  a  paper  by  W.  Townsend  Porter,^^  a  table 
which  shows  plainly  the  connection  between  the  number 
of  items  and  their  deviations  from  the  mean.  If  M  de- 
notes the  arithmetic  mean,  d  the  probable  error,  which  may 
vary  in  size  in  given  cases  according  to  the  precision  of 
the  individual  measurements,  then  1,000  items  will  be  dis- 
tributed about  the  mean  in  the  following  way : 

-f  nd  3 

-f4d 18 

4- 3d  67 

+  2d  162 

4-d  250" 

M 


—  d     250  ■ 

—  2d 162 

—  3d  67 

—  4d  18 

—  nd 3 


"  fluctuation "  for  the  square  of  the  modulus.  The  reciprocal 
value  of  the  modulus  is  the  "  precision,"  which  may  also  be  used 
as  a  measure  of  dispersion.     (See  footnote  37a,  p.  166.) 

'"  Cf.  especially  Bowley,  Elements  of  Statistics,  2nd  ed.,  pp.  281- 
292;  Feehner,  Kollekti\Tnasslehre,  pp.  18-22,  and  Duncker,  Die 
Methode  der  Variationsstatistik,  p.  36  f. 

^^  "On  the  Application  to  Individual  School  Children  of  the  Means 
Derived  from  Anthropological  Measurements  by  the  Greneralizing 
Method."  (Bull,  de  I'Inst.  intern,  de  Stat.,  Tome  VIII,  1895,  p. 
279  f.) 

**  That  one-quarter  of  the  cases  lie  within  the  probable  error  below 
the  average  and  one-quarter  above  the  average  is  a  direct  consequence 
of  the  definition  of  the  probable  error. 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      273 

The  greater  the  distance  from  the  mean  the  smaller  be- 
comes the  number  of  items  in  a  definite  proportion  as  de- 
fined by  the  Gaussian  probability  integral. 

A  similar,  but  more  detailed  table  for  the  distribution 
of  1,000  measurements  affected  with  accidental  errors  is 
given  by  Colajanni  in  his  Statistica  Teorica  (p.  192),  as 
follows : 


NUMBER  01"  DEVIATIONS 

SIZE  OP  THB  ERROR  (EXPRESSED 

BY  THE  PROBABLE  ERROR  AS  UNIT) 

From    0      to  0.5 

132 

132 

264.1 

"      0.5    "    1 

118 

li8 

235.9 

"       1       "    1.5 

94.2 

94.2 

188.3 

"      1.5    "    2 

67.2 

67.2 

134.3 

"       2       "    2.5 

42.8 

42.8 

85.6 

"       2.5    "    3 

24.4 

24.4 

48.7 

"      3       "    3.5 

12.4 

12.4 

24.8 

"      3.5    "    4 

5.6 

5.6 

11.2 

Over    4 

3.5 

3.5 

7.1 

500 

500 

1000 

The  attempt  has  frequently  been  made  to  explain  the 
characteristic  regular  conformation  of  the  series  resulting 
from  repeated  measurements  of  the  same  object.  It  is 
generally  supposed  that  this  conformation  can  be  traced 
back  to  the  presence  of  a  great  number  of  independent, 
equally  positive  and  negative  sources  of  error  (contribu- 
tory causes).  Every  one  of  these  sources  of  error,  taken 
by  itself,  can  cause  only  a  very  small  positive  or  negative 
error.  The  single  measurements,  however,  are  always  in- 
fluenced by  several  sources  of  error  simultaneously,  and 
that  in  various  combinations,  causing  greater  and  smaller 
deviations.  In  order  to  cause  a  greater  deviation,  either 
in  the  positive  or  the  negative  direction,  several  sources  of 
error  acting  in  the  same  direction  must  operate  simultane- 
ously or,  if  positive  and  negative  sources  of  error  occur 
together,  one  side  must  outweigh  the  other  considerably, 


274     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

since  positive  and  negative  sources  of  error  occurring  in 
equal  numbers  counterbalance  each  other.  Therefore, 
greater  deviations  from  the  average  (corresponding  to  the 
true  size  of  the  object  measured)  can  only  result  from  cer- 
tain combinations  of  the  sources  of  error,  which  combina- 
tions possess  probabilities  decreasing  with  the  sizes  of  the 
deviations.  From  this  results  the  fact,  which  originally  was 
ascertained  empirically,  that  in  a  series  of  repeated  measure- 
ments of  the  same  object  the  items  become  rarer  the  farther 
they  diverge  from  the  mean.-' 

It  is  not  always  easy  to  define  the  nature  of  the  sources 
of  error  in  a  given  case  of  concrete  measurements.  With 
physical  measurements  the  inadequacy  of  the  human  organs 
of  sense,  especially  of  the  eye,  the  uncertainty  of  the  hand, 
the  imperfections  of  the  instruments  used,  etc.,  may  be 
considered  to  be  the  sources  of  error. 

Series  which  obey  the  normal  law  of  errors  originate 
not  only  from  repeated  measurements  of  the  same  object, 
but  sometimes  from  single  (statistical)  observations  of 
distinct,  but  similar  items.  Therefore,  mathematical 
statisticians  usually  examine  statistical  series  in  order  to 
ascertain  whether  or  not  they  agree  with  the  normal  law 
of  error. 2*  If  there  is  sufficient  evidence  for  this,  then  the 
series  is  "  typical,"  i.  e.,  a  definite  normal  value  is  expressed 
in  the  series  with  merely  accidental  deviations.  The  math- 
ematical theorems  formulated  originally  for  repeated  meas- 
urements of  the  same  object  can  then  be  applied  with  good 
reason  to  the  statistical  series  in  question.^^     Especially  the 

''  Only  an  immaterial  modification  of  the  above  hypothesis  is  found 
if  we  follow  Pearson  and  do  not  proceed  from  the  assumption  that 
equal  numbers  of  positive  and  negative  sources  of  error  (contribu- 
tory causes)  are  present,  but  from  the  assumption  that  every  source 
of  error  acts  positively  or  negatively  with  the  same  probability. 

"  Cf.  for  the  methods  which  may  be  applied  to  this  investigation, 
Czuber,  Wahrscheinlichkeitsrechnung,  pp.  335-341. 

''With  Lexis  we  may  well  speak  in  this  case  of  a  "physical" 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      275 

dispersion  of  the  series  can  then  be  measured  and  expressed 
according  to  the  theory  of  errors  of  observation  by  a  single 
measure  of  dispersion,  such  as  the  error  of  mean  square, 
the  probable  error,  the  average  error,  the  "  modulus," 
or  the  "  precision."  The  errors  are  measured  from  the 
arithmetic  mean  of  the  series  which  may  be  called  the 
"  typical  "  mean.  But  since  this  mean  is  located  centrally 
and  since  the  items  are  densest  about  it,  the  arithmetic 
mean,  mode,  and  median  coincide,  or  differ  slightly  on 
account  of  the  small  number  of  observations.  If  the 
coincidence  of  the  series  with  the  normal  law  of  error  is 
established  and  if  the  average  and  the  average  dispersion 
of  the  series  are  known,  then  the  series  is  completely 
characterized  for  the  mathematical  statistician.  He  knows 
the  law  of  distribution  to  which  the  series  corresponds,  and 
the  constants  applying  to  the  individual  series. 

It  may  also  occur  that,  although  the  entire  series  does 
not  coincide  with  the  law  of  error,  a  portion  of  it  cor- 
responds to  this  law  and  shows  a  typical  conformation.  In 
such  a  case  the  typical  mean  is  not  expressed  by  the  average 
of  the  entire  series,  but  by  that  value  about  which  that 
part  of  the  series  which  agrees  with  the  law  of  errors  is 
distributed.  For  the  rest,  analogous  consequences  are 
found  as  in  the  case  of  wholly  typical  series. 

The  analogy  between  ''  typical  "  statistical  series  and 
series  of  repeated  measurements  of  the  same  object  also 
leads  to  a  plausible  explanation  of  the  former.  For  it  may 
be  conceived  that  typical  statistical  series  originate  from 
the  action  of  a  great  number  of  independent  causes  work- 
ing, either  in  positive  or  negative  direction  and  in  various 
combinations,  upon  the  individuals  or  items  observed  and 
expressed  in  the  series.  Thus,  the  conformations  of  the 
series  of  measurements  of  human  heights  which,  as  we 
know,   frequently   correspond  to  the  symmetrical  law  of 

method,  since  a  method  used  primarily  in  physical  observations  is 
applied  here  in  statistical  material. 


276     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

error,  may  be  traced  back  to  the  fact  that  numerous  in- 
fluences (such  as  different  nutrition,  various  modes  of  living 
in  adolescence,  atavism,  etc.),  partly  promoting  growth, 
partly  retarding  it,  work  upon  the  various  individuals  in 
varying  combinations,  thus  causing  a  symmetrical  distribu- 
tion about  the  average  height,  corresponding  to  the  law  of 
error 


26 


However,  "  typical  "  statistical  series  corresponding  to 
the  law  of  error  occur  but  rarely.  Quetelet  has  proved 
merely  that  they  exist  in  the  field  of  anthropometry.  He 
found  a  symmetric  distribution  about  the  average,  cor- 
responding to  the  law  of  chance,  especially  in  series  of 
measurements  of  height  and  girth  of  chest.  In  his  in- 
vestigations Quetelet  did  not  use  the  Gaussian  law,  but 
the  binomial  formula,  in  which  the  probabilities  of  the 
various  deviations  from  the  average  correspond  to  the  co- 
efficients of  the  binomial  expansion. ^^  Quetelet 's  expecta- 
tion that  series  corresponding  to  the  symmetrical  law  of 
chance  would  frequently  be  met  with  outside  of  the  field 
of  anthropometric  statistics  has  not  materialized.  We  know 
to-day  that  the  normal  law  of  error  cannot  be  taken  at  all 
as  the  general  law  of  distribution  of  statistical  phenomena. 
Statistical  series  or  part-series  corresponding  to  this  law  of 
distribution  are  found  only  in  very  rare  cases  outside  of 
the  anthropometric  field.  The  best  known  of  these  cases, 
discovered  by  Lexis,  is  the  symmetrical  distribution  of  the 
items  of  the  mortality  table  about  the  "  normal  "  length 
of  life,  the  mode  of  the  series  of  lifetimes  given  in  the 
mortality  table.  But  this  symmetrical  distribution  ex- 
tends only  to  certain  age  classes  belonging  to  the  "  normal 
group  "  which  includes  the  normal  length  of  life.  This  is 
a  comparatively  small  part  of  the  total  series  of  lifetimes 

^'  Cf.  Lexis,  "  Anthropologie  und  Anthropometrie  "  in  Handw.  der 
Staatsw.,  2nd  ed.,  Vol.  I,  p.  389  f. 

"  Cf.  about  Quetelet's  binomial  table  in  the  work  quoted  in  note, 
p.  390  ff. 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      277 

which  in  its  entirety  does  not  correspond  at  all  to  the  law 
of  error,  a  fact  which  is  expressed  by  the  strong  divergence 
of  the  arithmetic  mean  from  the  mode  of  this  series.^^-^ 

But  even  in  the  field  of  anthropometry  the  normal  law  of 
error  cannot  be  taken  as  the  universal  law  of  distribution. 
Anthropometric  series  frequently  show  unsymmetrical  dis- 
tributions (for  instance,  measurements  of  weights),  some- 
times contain  several  points  of  concentration,  which  are 
caused  by  lack  of  homogeneity,  and  sometimes  show  no 
perceptible  point  of  concentration. 

Statistics,  therefore,  usually  has  to  do  with  series  which 
neither  in  whole  nor  in  part  (as  in  the  ease  of  the  normal 
length  of  life)  admit  the  application  of  the  methods 
of  the  theory  of  errors  of  observation  in  their  original 

*' Even  the  symmetry  of  the  "normal  group"  of  lifetimes  has 
already  been  attacked  by  Pearson  on  the  basis  of  English  material, 
in  "  Contributions  to  the  Mathematical  Theory  of  Evolution,"  II : 
"  Skew  Variations  in  Homogeneous  Material."  Philosophical  Trans- 
actions of  the  Royal  Society  of  London,  Vol.  CLXXXV,  1,  1895,  A., 
p.  407.)  New  investigations  of  the  distribution  of  the  items  about 
the  mode  (normal  value)  have  been  published  by  E.  Blaschke  in 
Vorlesungen  iiber  math.  Statistik  (p.  154  ff.).  These  investigations 
refer  to  various  mortality  tables,  especially  mortality  tables  of  in- 
surance companies,  invalidity  tables,  and  to  several  series  giving  the 
number  of  sick  days  according  to  age,  and  the  distribution  of  those 
marrying  according  to  age.  Blaschke  (in  substance  agreeing  with 
Lexis )  found  that  the  distribution  of  the  items  above  the  "  normal 
age"  (towards  the  older  age  classes)  approximately  corresponds,  in 
most  cases,  to  the  law  of  distribution  of  errors,  but  that  below  the 
normal  age  the  coincidence  covers  only  a  narrow  range. 

^*  A  remarkable  coincidence  with  the  theory  of  error  is  also  shown 
by  the  fluctuations  of  the  net  proceeds  of  various  cereal  crops  in  Ger- 
many. The  fluctuations  of  the  net  proceeds  were  computed  for  some 
successive  cereal  crops  by  means  of  the  probability  calculus  ("Die 
Sehwankungen,  etc.,"  by  Dr.  Alfred  Mitscherlich,  Supplement  No. 
VIII  of  the  Zeitschrift  fur  d.  ges.  Staatsw.,  Tubingen,  1903),  and  it 
was  shown  to  what  important  practical  results  the  application  of 
the  theory  of  error  may  lead  in  forming  precepts  for  agricultural 
management. 


278     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

form  and,  consequently,  cannot  be  characterized  sufficiently 
by  a  single  measure  of  dispersion. 

With  this  fact  in  mind  mathematical  statisticians  have 
tried  to  express  and  explain  mathematically  the  conforma- 
tion of  non-symmetrical  series  by  modifications  and  general- 
izations of  the  Gaussian  law.  They  have  made  it  their 
purpose  to  establish  laws  of  distribution  for  non-sym- 
metrical series  which  shall  enable  them  to  combine  the 
various  statistical  series  into  theoretically  defined  groups 
of  a  higher  order  on  the  basis  of  uniform  principles. 

The  reason  for  accounting  for  non-symmetrical  statistical 
series  by  the  use  of  some  appropriate  extension  of  the  law 
of  error  is  much  more  obvious  when  we  consider  how 
many  unsymmetrical  statistical  series  exhibit  only  little 
asymmetry,  while  their  structure  otherwise  closely  ap- 
proaches that  of  really  "  typical  "  series.  The  transition 
from  the  symmetrical  series  which  correspond  to  the  normal 
law  of  error,  to  the  undoubtedly  unsymmetrical  series  is 
indefinable,  since  complete  symmetry  never  occurs  on  ac- 
count of  the  usually  relatively  small  number  of  observa- 
tions. Therefore,  in  a  given  case  opinions  may  differ  as 
to  whether  the  asymmetry  of  a  definite  series  is  to  be  con- 
sidered unessential  and  caused  merely  by  the  insufficient 
number  of  observations,  or  as  essential,  so  that  an  unsym- 
metrical law  of  distribution  is  expressed  in  the  dispersion 
of  the  items.  From  the  unsymmetrical  but  regular  statis- 
tical series  innumerable  forms  of  transition  lead  to  con- 
formations which  appear  to  possess  no  regularity  and  which 
seem  to  exclude  a  theoretical  explanation.  However,  math- 
ematical statisticians  have  frequently  tried  to  bring  even 
such  series  under  a  generalized  law  of  error  and  have  en- 
deavored to  prove  that  this  universal  law  is  expressed  even 
in  these  series,  although  with  considerable  modification. 

The  main  representative  of  this  idea  is  Edgeworth.  In 
his  opinion  preference  must  always  be  given  in  the  mathe- 
matical presentation  of  statistical  series  to  such  formulae 


J 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      279 

as  have  a  certain  relation  to  the  normal  law  of  error  and 
merely  show  deviations  from  it.  The  series  described  by 
these  formulae  are  supposed  to  be  caused  by  certain  modi- 
fications of  the  conditions  under  which,  otherwise,  the  nor- 
mal law  of  error  would  originate.^*^  Edgeworth  does  not 
consider  it  to  be  the  most  important  point  that  the  statistical 
material  correspond  to  the  highest  possible  degree  with  the 
formula  chosen  to  present  it,  but  he  insists  that  the  chosen 
law  of  distribution  be  based  on  a  hypothesis  which  is 
plausible  a  priori,  and  that  this  law  offer  a  plausible 
explanation  for  the  distributions  under  examination.  All 
this  holds  in  Edgeworth 's  opinion  for  the  law  of  error  and, 
consequently,  the  examination  of  statistical  series  must 
start  from  it.  Therefore,  the  representation  of  statistical 
series  by  means  of  the  law  of  error  and  appropriate  exten- 
sions of  it,  even  if  the  theory  does  not  completely  agree 
with  the  material  of  observation,  is  more  valuable  than  the 
representation  by  means  of  an  empirical  formula  (analytic 
function)  which,  even  though  it  fits  the  material  perfectly, 
does  not  furnish  any  explanation  of  the  conformation  of 
the  series  at  hand.^^ 

In  the  first  place  we  may  try  to  decompose  unsymmetrical 
series  into  constituent  series  which  separately  correspond  to 
the  normal  law  of  error  (method  of  separation  and  un- 
symmetrical Gaussian  law)  or  to  reduce  them  hypothetically 
to  an  original  conformation  which  corresponds  to  this  law 
(method  of  translation).  If  we  succeed  in  doing  this,  the 
further  mathematical  treatment  is  accomplished  by  the 
methods  of  the  normal  theory  of  error.  To  this  first  group 
of  methods  are  opposed  those  methods  in  which  we  do  not 
stop   with  the  normal  theory  of   error  but  explain   dis- 

*"  On  the  "  Representation  of  Statistics  by  Mathematical  Formulae," 
Journ.  of  the  Roy.  Stat.  Soc,  1898,  p.  674. 

"  The  presentation  of  statistical  series  by  means  of  empirical  for- 
mulae is  discussed  only  in  Appendix  I,  since  this  manner  of  pre- 
sentation does  not  proceed  from  the  average. 


280     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

tributions  by  means  of  skew  curves  of  error  as  well  as  by 
means  of  other  generalizations  of  the  law  of  error  or  the 
binomial  law. 

The  method  of  separation.  The  idea  of  considering  mul- 
timodal curves  as  complex  curves  caused  by  addition  or 
subtraction  of  two  or  more  curves,  is  obvious  even  to  non- 
mathematical  statisticians.  In  this  way  the  two  vertices 
in  the  frequency  curve  of  the  heights  of  the  recruits  of 
certain  French  departments  have  been  traced  by  J.  Bertillon 
to  the  mixture  of  two  races  of  different  type  as  to  height. 
This  process  is  more  difficult  with  unimodal  curves  which 
are  supposed  to  be  made  up  of  two  superimposed  modes. 
Pearson,^^  especially,  has  taken  up  the  problem  of  decom- 
posing such  curves  in  two  constituent  curves  of  norinal 
dispersion.  He  did  not  limit  himself  to  unsymmetrical 
curves,  but  declared  that  even  the  decomposition  of  sym- 
metrical curves  was  sometimes  necessary.  Unsymmetrical 
as  well  as  symmetrical  curves  may  consist  of  more  homo- 
geneous constituents  of  normal  dispersion,  the  separation 
of  which  is  of  scientific  value.  Furthermore,  the  decom- 
position of  an  unsymmetrical  curve  which  does  not  consist 
of  heterogeneous  material,  may  be  expedient,  since  by  this 
decomposition  we  obtain  a  measure  of  the  irregularity  of 
the  curve  and,  if  we  are  working  in  the  field  of  biology  or 
anthropology,  a  measure  of  the  evolution  of  the  given 
character,  which  may  express  an  improvement  or  a  de- 
generation of  the  species. 

Pearson  has^^  also  promised  an  investigation  of  the 
problem  of  decomposing  complex  curves  into  skew  con- 
stituent curves  and,^^  by  way  of  experiment,  has  decomposed 

'''"Contributions  to  the  Mathematical  Theory  of  Evolution"  in 
Philosophical  Transactions  of  the  Roy.  Soc.  of  London,  Vol.  CLXXXV, 
1894,  A,  pp.  71-110.  See  also  Duncker,  Die  Methode  der  Variations- 
statistik,  p.  16  f. 

''  Philosophical  Transactions  of  the  Roy.  Soc.  of  London,  Vol. 
CLXXXVI,  1895,  Pt.  I,  A,  p.  406.  '*  Ibid.  pp.  406-410. 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      281 

the  series  of  lifetimes  given  in  the  English  mortality  table 
into  5  constituent  curves,  whose  modes  are  at  the  ages  of 
71.5  years,  41.5  years,  22.5  years,  3  years,  and  in  the  be- 
ginning of  the  first  year,  corresponding  respectively  to  the 
mortalities  of  old  age,  of  middle  age,  of  adolescence,  of 
childhood,  and  of  infanc}^  Of  these  five  constituent  curves, 
those  of  the  mortalities  of  old  age,  childhood,  and  of  infancy 
are  unsymmetrical,  those  of  middle  age  and  of  adolescence 
are  almost  symmetrical.  It  is  remarkable  that  Pearson 
has  found  an  unsymmetrical  distribution  for  the  mortality 
of  old  age,  at  least  on  the  basis  of  the  English  material, 
while  Lexis  has  ascertained  a  symmetrical  distribution  about 
the  maximum,  the  normal  length  of  life,  on  the  basis  of 
the  material  of  various  other  countries. 

The  decomposition  of  series  according  to  the  method  of 
separation  may  enable  us  to  draw  conclusions  of  great 
significance,  as  the  following  example  shows.  In  his  book 
Die  naturliclie  Auslese  heim  Menschen,  Ammon  has  based 
his  assertion  of  an  evolution  of  the  cephalic  index  of  the 
South  German  on  the  comparison  of  exhumed  Germanic 
skulls  with  modern  skulls.  In  opposition  to  this,  Pearson 
states  in  the  paper  quoted  above  that  he  has  succeeded  in 
decomposing  the  unsymmetric  curve  of  the  old  Germanic 
skulls  into  two  constituent  curves,  one  of  which  agrees  with 
the  curve  found  for  modern  skulls.  From  this,  Pearson 
concludes  that  the  older  skulls  come  from  a  mixed  popula- 
tion, and  that  an  evolution  of  the  cephalic  index  cannot  be 
proved  for  that  part  of  the  population  which  is  still  living. 

Pearson's  method  has  been  criticised  thoroughly  by  Edge- 
worth,^^  who  acknowledges  that  Pearson's  method  of  sepa- 
ration— in  spite  of  its  mathematical  complexity — is  very 
valuable,  especially  because  it  is  not  based  on  a  merely 
theoretical  hypothesis  but  on  actual  facts.  It  is,  for  in- 
stance, known  that  the  statistical  series  representing  the 

"  Journ.  of  the  Roy.  Stat.  Sec,  Vol.  LXII  (1S99),  p.  125  j  see  also 
Vol.  LXV  (1902),  p.  327. 


282     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

heights  of  the  Italian  population  can  be  divided  into 
constituent  series  for  the  various  provinces,  which  show 
averages  of  different  sizes  and  represent  different  types  of 
height. 

The  unsymmetrical  Gaussian  law.  An  unsymmetrical 
series  corresponds  to  this  law  if  the  two  parts  of  the  series 
located  on  either  side  of  the  mode  correspond  separately  to 
the  normal  Gaussian  law  and  if  the  non-symmetry  of  the 
series  is  caused  merely  by  the  fact  that  the  two  parts 
of  the  series  have  different  mean  errors  (or  probable  errors, 
moduli,  etc.,  according  to  the  measure  of  dispersion  used). 
According  to  the  unsymmetrical  Gaussian  law,  unsymmet- 
rical series  are  considered  to  be  complex  series  consisting  of 
two  half-normal  curves  of  different  precision  joined  at  their 
modes.  Obviously,  only  those  series  which  possess  but  one 
mode  may  be  so  considered.  If  a  series  is  found  for  which 
the  above  assumption  holds,  then  it  can  be  completely  de- 
fined mathematically  by  the  statement  of  the  mode  and  the 
mean  error  (or  any  other  measure  of  dispersion)  of  each 
of  the  two  constituent  series. 

Professor  Edgeworth,  who  has  thoroughly  explained  and 
discussed  the  method  of  the  non-symmetrical  Gaussian  law 
under  the  heading  "Method  of  Composition,"^®  admits 
that,  as  a  matter  of  fact,  this  method  can  sometimes  be 
used  with  success  in  the  mathematical  treatment  of  certain 
series.  But  he  also  states  that  very  unsymmetrical  series 
cannot  easily  be  treated  by  means  of  this  method  because 
a  curve  of  regular  conformation  cannot  originate  from 
the  halves  of  two  normal  curves  of  considerably  varying 
dispersion.  Edgeworth  also  objects  to  the  "  method  of 
composition  "  because  it  contains  no  plausible  reason  for 
the  assumed  kind  of  conformation  of  the  series.  Why  the 
constituent  series  on  either  side  of  the  mode  should  contain 
different  mean  errors,  is  not  explained  at  all  by  the  assump- 

"  Journ.  of  the  Eoy,  Stat.  Soc,  Vol.  LXII  (1899),  pp.  373-385  and 
p.  543. 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      283 

tions  used  in  this  method,  and  it  is  not  clear  Avhy  such  an 
artificial  and  manufactured  conformation  of  the  series 
should  originate. 

The  method  of  the  unsymmetrical  Gaussian  curve  has  re- 
ceived the  most  thorough  treatment  by  G.  Th.  Fechner. 
In  his  paper  published  in  1878,  **  Uber  den  Ausgangswert 
der  kleinsten  Abweichungssumme,"  Fechner  originally 
presented  the  idea  of  considering  unsymmetrical  series  of 
measurements  as  complex  curves  consisting  of  halves  of 
normal  curves  of  errors  of  different  precisions,  and  he  has 
developed  the  idea  most  exhaustively  in  his  Kollektivmass- 
lehre  (1897).  In  this  work  Fechner  has  given  a  number 
of  series  which  agree  with  the  unsymmetrical  Gaussian  law 
and  which,  on  the  basis  of  his  investigations,  justify  the 
assumption  that  this  law  must  be  considered  to  be  the 
universal  law  of  distribution  of  "  collective  objects  " — a 
term  which  Fechner  uses  to  designate  all  accidentally  vary- 
ing objects,  especially  anthropological,  biological,  and  mete- 
orological measurements.  It  is,  however,  limited  to  series 
with  relatively  weak  fluctuations  about  the  mode,  such  as 
appear  in  most  collective  objects. 

In  the  case  of  great  asymmetry  and  great  relative  devia- 
tions Fechner  has  recommended  the  "  logarithmic  "  treat- 
ment of  the  given  series.^^  This  method  consists  in  finding 
the  logarithms  of  the  items,  determining  the  mode-  in  the 
series  of  these  logarithms,  and  examining  the  deviations 
of  the  individual  logarithms  from  their  mode.  By  this 
method  Fechner  has  arrived  at  a  logarithmic  generalization 
of  the  unsymmetrical  Gaussian  law,  inasmuch  as  he  found 
that  this  law  of  distribution  holds  also  for  the  series  of 
logarithms  which  he  examined. 

According  to  Fechner,  the  logarithmic  treatment  is  valua- 
ble principally  because,  by  translating  the  logarithmic  mode 
and  the  logarithmic  deviations  into  the  appertaining  natural 

"  Cf.  Kollektivmasslehre,  pp.  77-83  and  pp.  339-351,  and  "Aus- 
gangswert, etc.,"  pp.  14-17. 


284     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

numbers,  we  obtain  the  relative  deviations  and  their  base 
value,  the  relative  mode.  The  latter  differs  from  the  arith- 
metic mode.  The  relative  deviation  indicates  the  percent 
of  the  base  value  that  the  item  is  above  or  below  it,  while 
the  usual  arithmetic  deviation  indicates  the  absolute  differ- 
ence between  the  item  and  the  mean.  According  to 
Fechner,  the  relative  deviations  have  a  special  significance 
in  the  field  of  collective  objects — but  not  in  physical  and 
astronomical  observations — since  "  the  variation  of  an  ob- 
ject bears  a  certain  relation  to  the  size  of  the  object 
itself,  so  that  the  variation  depends  essentially,  although 
not  exclusively,  upon  the  size  of  the  object;  according  to 
this,  the  height  of  a  blade  of  grass,  taken  absolutely,  varies 
less  than  that  of  a  fir  tree,  but  we  cannot  assert  that 
relatively  it  varies  less. ' '  ^^  Therefore,  according  to  Fech- 
ner, the  variation  of  a  collective  object  can  be  judged 
better  from  relative  deviations  than  from  arithmetic  de- 
viations. Furthermore,  the  fact  that  arithmetic  deviations 
are  limited,  inasmuch  as  an  object  cannot  decrease  more 
than  its  own  size,  argues  for  the  logarithmic  treatment 
and  the  computation  of  relative  deviations,  because  this 
limitation  is  removed  when  referring  to  logarithmic  and  the 
ensuing  relative  deviations,  since  any  object  may  decrease  as 
well  as  increase  in  infinite  ratio.^^ 

In  Fechner 's  opinion  the  normal  (symmetric)  Gaussian 
law  based  on  arithmetic  deviations  is  merely  a  special  case  of 
the  unsymmetric  Gaussian  law  also  based  on  arithmetic  de- 
viations. It  corresponds  to  the  limiting  case  where  the 
deviations  on  both  sides  of  the  mode  are  equal.  Among  the 
infinitely  many  degrees  of  varying  asymmetry  the  case  of 
the  complete  disappearance  of  asymmetry  possesses  only 
little  probability.  On  the  other  hand,  the  logarithmic  law 
of  distribution,  which  is  to  be  applied  to  collective  objects 

** "  Ausgangswert,  etc.,"  p.  14;  see  also  Kollektivmasslehre,  p. 
78  f.;  and  above,  p.  262  f. 

**  "  Ausgangswert,  etc.,"  p.  16,  and  Kollektivmasslehre,  p.  77. 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      285 

with  strong  relative  fluctuations,  agrees  remarkably,  accord- 
ing to  Fechner,  with  the  unsymmetrical  Gaussian  law  based 
on  arithmetic  deviations  if  there  are  only  weak  relative 
fluctuations,  and  becomes  identical  with  it  as  the  fluctua- 
tions decrease,  so  that  the  logarithmic  law  may  be  taken 
as  the  most  general  law  of  distribution  of  collective  objects. 

Fechner  has  also  tried  to  explain  the  origin  of  unsym- 
metrical series  which  correspond  to  the  unsymmetrical 
Gaussian  law.*°  He  assumes  the  existence  of  an  indefinite 
number  of  forces  or  special  conditions  which,  independent 
of  each  other,  exert  a  modifying  influence  upon  the  sizes 
of  the  specimens  of  a  collective  object.  According  to 
Fechner 's  hypothesis,  every  force,  when  active,  causes  so 
small  a  change  that  the  second  power  of  the  latter  may  be 
neglected  in  comparison  with  finite  values.  There  is  the 
probability  p  for  the  presence  of  an  effect,  and  the  proba- 
bility q  =  1  —  p  for  the  absence  of  an  effect  of  the  activity 
©f  each  individual  force.  On  the  basis  of  this  hypothesis 
an  unsymmetrical  distribution  is  generally  obtained;  a 
symmetrical  distribution  corresponding  to  the  normal  Gaus- 
sian law  originates  merely  in  the  special  case,  when  p 
equals  q. 

The  metJiod  of  translation.  Edgeworth,  the  originator 
of  this  method,  proceeds  from  the  following  observation :  *^ 
If  a  series  is  given  which  corresponds  to  the  normal  sym- 
metrical law  of  error  and  if,  based  on  this  series,  a  second 
series  be  computed  the  items  of  which  represent  assigned 
functions  of  the  items  of  the  first  series,  then  the  second, 
generated  series,  under  certain  conditions,  has  an  unsym- 
metrical conformation  which  is  mathematically  predeter- 
mined. Let  the  original  series  be  the  distribution,  corre- 
sponding to  the  normal  law  of  error,  of  the  heights  of  the 
males  of  any  nation.  Now,  if  a  second  series  is  formed 
from,  say,  the  squares  of  those  values  in  which  the  heights 

*°  Kollektivmasslehre,  pp.  306-320  and  p.  357. 

"  Cf.  Journ.  of  the  Koy.  Stat.  Soc,  1898,  p.  675,  and  1899,  p.  537. 


286     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

of  the  first  series  result  when  taken  from  a  definite  point, — 
for  instance,  from  the  height  of  the  shortest  man, — ^then  this 
new  generated  series  shows  an  unsymmetrical  conforma- 
tion. According  to  Edgeworth,  the  essence  of  the  method 
of  translation  is  to  treat  unsymmetrical  series  of  measure- 
ments as  though  every  item  were  an  assigned  function  of 
an  item  of  a  normal  series  which  is  to  be  ascertained.  The 
problem,  then,  is:  to  find  the  generating  normal  curve, 
which  may  be  determined  by  computing  the  average  and 
the  modulus.  An  additional  problem  is:  to  determine  a 
point  such  that  its  distance  from  each  point  of  the  generat- 
ing curve  is  in  functional  relation  with  a  corresponding 
point  of  the  generated  curve.  With  the  help  of  some 
examples  taken  from  the  field  of  meteorology  (especially 
unsymmetrical  series  of  barometric  measurements)  Edge- 
worth  has  shown  how  this  method  can  be  applied  in  practice. 
The  method  of  the  skew  curves  of  error.  The  aim  of 
the  methods  mentioned  thus  far  is  to  divide  unsymmetrical 
series  into  symmetrical  series  or  constituent  series  or  to 
reduce  unsymmetrical  series  to  symmetrical  series  in  order 
to  apply  a  thorough  mathematical  investigation  to  the  sym- 
metrical series  or  constituent  series  thus  obtained.  Instead 
of  proceeding  in  this  way  we  may  proceed  with  a  direct 
mathematical  treatment  of  the  unsymmetrical  series.  The 
problem,  then,  is  to  find  a  skew  curve  of  error  with  which 
the  actual  material  of  the  series  agrees  sufficiently  so  that 
this  curve  or  the  corresponding  algebraic  function  may  be 
used  in  expressing  the  series  in  question.  Of  course  stating 
an  average  of  the  deviations  from  the  arithmetic  mean  of 
the  series  is  obviously  not  sufficient  to  completely  character- 
ize a  skew  curve  of  error,  but  a  numerical  expression  of 
the  degree  of  asymmetry  must  also  be  computed.^^     How- 

*^  Cf.  Edgeworth,  on  the  "Asymmetrical  Probability-Curve,"  Philo- 
sophical Magazine,  1896;  and  Journ.  of  the  Roy.  Stat.  Soc,  1900, 
p.  76;  and  Bowley,  Elem.  of  Stat.,  2nd  ed.,  Appendix,  pp.  329-334; 
and  Journ.  of  the  Roy.  Stat.  Soc,  1902,  pp.  331-354. 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      287 

ever,  only  series  which  do  not  deviate  too  much  from  the 
normal  form  can  be  reproduced  by  means  of  skew  curves 
of  error.*^ 

The  generalized  method  of  translation.  Another  method 
belonging  in  this  list  is  the  generalized  method  of  transla- 
tion, developed  by  Professor  Edgeworth.^*  Its  most  general 
application  is  to  relate  a  given  series  to  an  unsymmetrieal 
generating  curve.  By  means  of  this  method  Edgeworth 
has  succeeded  in  expressing  mathematically  even  series  of 
strongly  unsymmetrieal  conformation,  as  well  as  unilateral 
curves  the  modes  of  which  are  in  the  beginning  or  at  the 
end  of  the  series.  Such  unilateral  curves  frequently  occur 
in  botanical  statistics.  But  they  also  occur  in  population 
and  economic  statistics.  Such  series  are,  principally,  the 
childhood  mortality  which  begins  with  a  maximum,  and 
the  series  of  taxable  incomes,  in  which  the  lowest  classes 
are  of  the  greatest  frequencies,  at  least  in  some  countries, 
Edgeworth  admits  that  such  unilateral  series,  especially 
the  distribution  of  incomes,  apparently  have  nothing  to 
do  with  the  law  of  error.*^  But  the  income  may  be  in 
functional  relation  to  some  other  criterion  which  is  governed 
by  the  law  of  accidental  deviations,  such  as  individual 
ability.*^     Therefore,  Edgeworth  thinks  it  is  admissible  to 

*'  Especially  Thiele,  Bruns,  Charlier,  and  Kapteyn  have  worked 
on  the  skew  curves  of  error  from  a  theoretical  mathematical  point 
of  view. 

"  Cf.  Journ.  of  the  Roy.  Stat.  Soc,  1899,  p.  537. 

*'  The  attempts  made  outside  of  the  field  of  theory  of  error  to 
characterize  such  series  by  means  of  analytical  formulce  (which  do 
not  originate  from  the  averages  of  the  series  in  question)  are  dis- 
cussed in  Appendix  I   (A). 

*"  According  to  the  curve  drawn  by  Galton,  the  various  degrees 
of  human  faculty  are  distributed  about  the  mean  according  to  the 
Gaussian  law.  (Cf.  especially  Hereditary  Genius  and  Inquiries  into 
Human  Faculty  and  Its  Development.)  Cf.  also  Ammon's  discus- 
sions (Die  Gesellschaftsordnung  und  ihre  natiirlichen  Grundlagen) 
on  the  relations  between  the  distribution  of  incomes  and  the  curve 
of  faculties. 


288     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

treat  the  distribution  of  incomes  and  similar  unilateral 
series  by  means  of  a  generalized  method  of  translation,  and 
believes  himself  to  have  obtained  in  this  way  better  results, 
i.  e.,  a  better  agreement  between  theory  and  material  of 
observation,  than  Pearson,  who  also  has  treated  series  of 
this  kind  by  means  of  a  special  generalization  of  the  law 
of  error  (discussed  later). 

The  "  generalized  law  of  error  "  of  Edgeworth.  The  law 
of  error  has  recently  been  given  its  most  general  form 
by  Edgeworth,  whose  "  generalized  law  of  error  "  or  "  ex- 
ponential law  of  great  numbers  "  *'^  is  adapted  to  explain 
the  widest  range  of  statistical  series.  The  normal  law  of 
error  as  well  as  the  method  of  translation  are  only  special 
cases  of  this  generalized  law,  the  validity  of  which  can 
therefore  be  proved  in  many  fields  of  nature  and  social 
life  when  they  are  treated  statistically. 

Pearson's  generalized  probahility  curve.  Finally,  the 
generalized  probability  curve  must  be  mentioned.  It  was 
formulated  by  Pearson  and  he  has  illustrated  it  by  numer- 
ous examples  taken  from  various  fields.*^ 

It  corresponds  to  the  general  binomial  (rr^+rrr-)°  in  a 
similar  way  as  the  normal  curve  (Gaussian  curve  of  error) 
does  with  the  definite  binomial  (%  -}-'  ^2)°.*^  It  may  there- 
fore, be  symmetrical  or  unsymmetrical  and  of  unlimited 

*"<  Cf.  Journ.  of  the  Roy.  Stat.  See,  1906,  p.  497  ff. 

*'  See  "  Contributions  to  the  Mathematical  Theory  of  Evolution," 
II:  "Skew  Variation  in  Homogeneous  Material"  (Philosophical 
Transactions  of  the  Royal  Society  of  London,  Vol.  CLXXXVI,  Pt.  I 
(1895),  A.,  pp.  343-414)  and  divers  articles  in  Biometrika,  a  journal 
for  the  statistical  study  of  biological  problems,  edited,  in  consultation 
with  Francis  Galton,  by  F.  R.  Weldon,  Karl  Pearson,  and  C.  R.  Daven- 
port. (Cambridge,  since  October,  1901.)  Cf.  also  C.  B.  Davenport, 
Statistical  Methods  with  Special  Reference  to  Biological  Varia- 
tion, New  York,  1904,  and  W.  P.  Elderton,  Frequency-Curves  and 
Correlation,  London,  1906. 

"  Pearson,  loc.  cit.,  p.  345,  and  Duncker,  Die  Methode  der 
Variationsstatistik,  p.  14. 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      289 

extension  on  both  sides  of  the  mean  on  the  axis  of  the 
abscissas,  or  limited  on  one  side  or  on  both  sides  of  the 
mean.     Thus,  the  following  five  types  are  found : 

Type  1.  Axis  of  the  abscissas  limited  on  both  sides,  curve 
unsymmetrical. 

Type  2,  Axis  of  the  abscissas  limited  on  both  sides, 
curve  symmetrical. 

Type  3.  Axis  of  the  abscissas  limited  on  one  side,  curve 
unsymmetrical. 

Type  4.  Axis  of  the  abscissas  unlimited  on  both  sides, 
curve  unsymmetrical. 

Type  5.  Axis  of  the  abscissas  unlimited  on  both  sides, 
curve  symmetrical.  Type  5  is  the  normal  curve  or  Gaus- 
sian curve  of  error.^" 

An  unsymmetrical  conformation  of  a  given  curve  can  be 
explained,  according  to  Pearson,  in  that  not  all  the  con- 
tributory causes  bring  about  equally  great  positive  or  nega- 
tive deviations  with  the  same  probability,  which  is,  accord- 
ing to  Pearson,  to  be  assumed  for  the  origin  of  a  normal 
curve  of  error.^^  Furthermore,  Pearson's  probability  curve 
is,  in  its  most  general  form,  based  on  the  assumption  that 
the  contributory  causes  are  not  independent  of  each  other.^^ 

""Pearson,  loc.  cit.,  p.  360;  Duncker,  loc.  cit.,  p.  15  f.  (Cf. 
the  classification  of  curves  given  by  W.  Palin  Elderton  in  Frequency- 
Curves  and  Correlation. — Translator.) 

°^  If  we  do  not  proceed  like  Pearson,  when  explaining  the  sym- 
metrical curve  of  error,  from  the  assumption  that  the  contributory 
causes  may  with  equal  probability  cause  equally  great  positive  and 
negative  deviations,  but  from  the  more  usual  assumption  that  a 
symmetrical  curve  of  error  must  be  based  on  the  activity  of  two 
equally  strong  groups  of  contributory  causes,  one  of  them  causing 
positive  and  the  other  negative  deviations,  then  we  may  explain 
the  origin  of  unsymmetrical  curves  of  error  by  the  assumption  that 
the  contributory  causes  acting  positively  and  negatively  in  producing 
individual  variation  are  not  present  in  equal  numbers.  (Cf.  Duncker, 
Die  Methode  der  Variationsstatistik,  p.  33.) 

°-  Cf.  in  this  connection  the  criticism  by  Edgeworth  in  Jour,  of  the 
Roy.  Stat.  Soc,  1899,  p.  635  f. 


290     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

Great  stress  is  put  by  Pearson  on  the  fact  that  the  range 
of  variation  of  most  phenomena  is  limited,  owing  to  their 
nature,  a  fact  which  is  theoretically  opposed  to  the  applica- 
tion of  the  normal  curve  of  error  which  is  unlimited  on 
both  sides.  Thus,  every  age  distribution  has  a  fixed  inferior 
limit  and,  usually,  also  a  superior  limit;  for  instance,  the 
age  distribution  of  the  women  who  give  birth  to  children 
during  the  year  is  limited  on  one  side  by  the  age  of 
puberty,  on  the  other  side  by  the  climacteric  age.^^ 

The  practical  application  of  Pearson's  theory  consists, 
above  all,  in  computing  the  function  to  which  a  concrete 
statistical  series  corresponds.  In  addition,  the  mean  of 
the  series,  the  standard  deviation,  the  number  of  observa- 
tions contained  in  the  series,  and  a  measure  for  the  degree 
of  agreement  between  observation  and  theory  are  needed 
to  characterize  the  series.  "  With  some  practice  these  few 
data  give  the  reader  such  a  clear  and  complete  picture 
of  the  manner  of  distribution  as  cannot  be  obtained  from 
a  word  description,  however  complete."^* 

Pearson  has  shown  by  illustrations  that  numerous  statis- 
tical series  from  the  fields  of  meteorology,  biology,  and 
anthropology,  as  well  as  from  the  fields  of  demography  and 
economics,  may  be  presented  according  to  his  method.  Not 
only  has  he  treated  series  which  are  distributed  about  a 
mode  with  decided  asymmetry — such  as  barometric  meas- 
urements— but  he  has  also  proved  that  even  in  series  of 

"'  Pearson,   loc.   cit.,   p.   359. 

"  Duncker,  loc.  cit.,  p.  33.  The  variations  of  Miillerian  glands 
of  hogs  are  characterized  by  applying  Pearson's  method  in  the  fol- 
lowing way:  M  (mean)  =3.5010,  e  ( standard  deviation )  =1.6808,  A 
(degree  of  coincidence  between  observation  and  theory,  that  is 
measure  of  the  lack  of  coincidence  between  the  empirical  and  the 
computed  variation  polygon  according  to  the  manner  of  computation 
proposed  by  Duncker)  =1.57^,  n  (number  of  observations)  =2000, 
formula  of  curve: 

Ar,r,(,^  (.    .          X        \     4.8434    /,  X         \  17.6189 

y=473.9  X  (l  +  -j;^^-)  (l 15:6023-) 


QUANTITATIVE  INDIVIDUAL  OBSERVATIONS      291 

apparent  symmetrical  distribution — such  as  series  of  meas- 
urements of  heights — a  better  agreement  between  the  theory 
and  the  observations  can  be  obtained  by  assuming  a  certain 
asymmetry  than  by  basing  the  series  on  a  symmetrical 
curve.^^  He  also  examined  series  of  extreme  asymmetry, 
so-called  unilateral  series  which  start  with  the  mode,  and 
presented  them  as  a  special  case  under  his  generalized  law 
of  probability.  He  used  as  illustrations  of  this  kind  of 
series :  the  distribution  of  the  houses  of  England  according 
to  value,  a  distribution  characterized  by  the  fact  that  the 
houses  of  the  lowest  value  are  the  most  frequent,  and 
several  botanical  series,  for  instance,  the  number  of  blossoms 
and  petals  of  plants  of  a  definite  species  in  which  the  lowest 
values  are  the  most  frequent,  while  greater  values  occur 
with  decreasing  frequency. 

Pearson  found  that  the  types  1  and  4  occur  most  fre- 
quently; botanical  measurements  usually  correspond  to 
type  1,  zoological  to  type  4."^ 

"  As  opposed  to  this.  Lexis'  opinion  may  be  mentioned  "  that  the 
proof  of  even  an  approximate  symmetry  is  of  greater  theoretical 
significance  than  the  accurate  presentation  of  a  distribution  by  an 
unsymmetrical  curve  the  origin  of  which  cannot  be  conceived  as 
plainly  as  that  of  the  normal  curve."  (Article,  "Anthropologic 
und  Anthropometric  "  in  Handw.  d.  Staatsw.,  2nd  ed.,  Vol.  I,  p.  397.) 

°°a  The  fundamental  distinctions  between  the  two  schools  led, 
respectively,  by  Pearson  and  Edgeworth  are  stated  as  follows  by 
A.  L.  Bowley:  "In  the  one,  the  predominate  idea  is  to  find  a  purely 
empirical  formula  to  fit  the  observations,  the  excellence  of  the 
formula  being  measured  by  the  closeness  of  the  fit  and  the  fe^^Tiess 
of  the  arbitrary  constants,  and  then  to  assume  that  the  observations 
can  be  replaced  by  the  mathematical  formula,  and  the  laws  of 
chance  applied;  in  the  other  it  is  rather  sought  to  find  a  priori 
what  mathematical  law  will  tend  to  be  obeyed  if  certain  postulates 
are  given  as  to  the  genesis  of  the  obsers'ed  quantities,  and  to  dis- 
cover those  postulates  which  yield  a  law  that  adequately  describes 
the  phenomena.  The  methods  and  the  formulae  of  the  two  schools 
are  intimately  connected,  and  in  many  cases  yield  identical  results." 
(Journ.  Roy.  Stat.  Soc,  Vol.  LXIX,  p.  747.)— Tbanslatob. 


CHAPTER  III 

THE  DISPERSION  OF  SERIES  COMPOSED  OF  MAGNI- 
TUDES LIMITED  IN  A  DEFINITE  WAY  (CONSTITU- 
ENTS OF  A  GREATER  TOTALITY) 

Series  whose  items  denote  the  sizes  of  masses  limited 
in  a  definite  way  (constituents  of  a  larger  totality)  form 
the  second  group  of  series  distinguished  in  the  beginning 
of  the  book.  The  series  of  this  group  are  time,  space, 
qualitative,  or  quantitative  series  depending  on  the  criterion 
according  to  which  the  items  are  selected.  For  example, 
to  this  group  belong  those  series  the  items  of  which  indicate 
how  many  births  and  deaths  have  occurred  in  the  single 
months  of  the  year,  or  in  successive  years  of  a  longer  period, 
or  how  many  inhabitants  are  in  the  different  districts  of 
a  country,  or  how  many  persons  follow  the  various  occupa- 
tions. The  series  of  the  second  group — as  has  already  been 
explained — admit  only  the  computation  of  the  arithmetic 
average  of  the  items  (constituent  masses)  of  the  series,  but 
not  the  computation  of  other  means. 

First  of  all,  the  average  of  the  series  may  be  supple- 
mented by  the  statement  of  the  extreme  values,  the  maxi- 
mum and  the  minimum.  In  doing  this  we  may  limit  our- 
selves to  indicating  the  highest  and  the  lowest  figures  occur- 
ring in  the  series.  But  we  may  also  describe  more  exactly 
the  items  to  which  these  extreme  values  refer,  i.  e.,  the 
years  in  which  maximum  and  minimum  occurred,  or  the 
districts  which  have  the  greatest  or  the  smallest  number 
of  population,  etc.^^ 

■^  Ottingen  speaks  of  the  "tenacity"  of  a  time  series  with  small 
differences  between  the  average  and  the  extreme  values;  in  case  of 

393 


DEFINITELY  LIMITED  MAGNITUDES  293 

If  the  statement  of  the  extreme  values  does  not  seem  to 
be  sufficient,  then  we  may  resort  to  the  device,  also  used 
with  series  of  quantitative  individual  observations,  of  pre- 
senting certain  classes  which  are  of  significance  in  the  given 
case. 

Finally,  in  order  to  obtain  a  uniform  expression  for  the 
dispersion  of  the  series,  the  average  deviation  may  be 
computed  as  a  supplement  of  the  average.  This  compu- 
tation is  effected  by  combining  the  deviations  of  all  the 
items  from  the  average  and  the  result  may  be  stated 
either  in  absolute  size  or  in  per  cent  of  the  average.  Differ- 
ent statistical  writers,  especially  Ad.  Wagner  ^^  and  G.  von 
Mayr,^^  have  frequently  used  this  method  in  time  series. 
Such  measures  of  dispersion,  however,  give  a  clear  picture 
of  the  deviations  in  the  series  under  consideration  as  well 
as  in  series  of  quantitative  individual  observations  only  if 
the  items  be  distributed  symmetrically  on  both  sides  of  the 
average. 

large  differences,  however,  he  speaks  of  the  "  sensibility "  of  the 
series. 

°'  Cf.  Gesetzmassigkeit  in  den  scheinbar  willkiirlichen  menschlichen 
Handlungen,  "Comparative  Suicide  Statistics  of  Europe"  (1864), 
for  instance  pp.  88  and  93.  In  the  latter  place  Wagner  has  given 
the  absolute  size  of  the  average  deviation  as  well  as  its  percentage 
of  the  mean. 

""  Cf.  Gesetzmassigkeit  im  Gesellschaftsleben  (1877),  p.  57  ff.  In 
"  Statistik  der  Bettler  und  Vaganten  im  Konigreiche  Bayern " 
(1865),  V.  Mayr  has  merely  added  the  deviations  (of  the  numbers 
of  poor  for  1835-1860)  from  the  average  number  of  poor  for  this 
period,  and  has  used  this  sum  as  measure  of  the  deviations  without 
computing  a  real  mean  deviation  (by  dividing  the  sum  of 
the  deviations  by  their  number).  (Cf.  ibid.  pp.  19  and  28.) 
Also,  Engel  occasionally  used  indices  of  fluctuation;  thus  in 
his  Bewegung  der  Bevolkerung  in  Konigreich  Sachsen  in  den 
Jahren  1834-1850  he  computed  the  mean  duration  of  marriage  by 
dividing  the  number  of  existing  marriages  by  the  number  of  wed- 
dings, correcting  the  latter  by  a  coefficient  computed  from  the  mean 
annual  fluctuation  of  the  frequency  of  marriages. 


294     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

The  time  series  consisting  of  absolute  numbers,  which 
form  the  most  important  subdivision  of  the  group  of  series 
under  consideration,  however,  very  frequently  show  a  char- 
acteristic conformation  which  does  not  lie  in  the  distribu- 
tion of  the  items  about  the  mean  but  in  some  other  regu- 
larity— for  instance,  a  direction  of  development  progressing 
in  a  definite  manner  and  keeping  pace  with  time,  or  a 
definite  periodicity.  If  such  a  conformation  is  present, 
then  the  measurement  of  the  dispersion  of  the  items  about 
the  mean  is  not  sufficient.  In  such  a  case  the  actual  char- 
acteristic property  of  the  series,  for  instance,  the  direction 
of  its  development  or  its  periodicity,  cannot  be  seen  from 
the  size  of  the  measure  of  dispersion.  In  order  to  ascertain 
this  property,  the  items  must  be  studied  successively  and  be 
presented  by  means  of  other  methods  discussed  in  Appendix 
I,  because  when  measuring  the  dispersion  we  do  not  take  the 
succession  of  the  items  into  consideration  but  merely  meas- 
ure their  distances  from  the  average.^^ 

The  fact  that  statistical  series  with  only  slight  time 
fluctuations  have  frequently  been  found,  has  caused  many 
discussions  in  statistical  literature.  On  the  basis  of  the 
observation  of  the  invariability,  constancy,  or  steadiness  of 
certain  series,  statistical  ''  laws  "  were  constructed,  fre- 
quently without  adequate  basis,  as  in  the  case  of  Quete- 
let's  budget  of  penitentiaries  and  scaffolds.  Likewise,  im- 
portant   philosophical    conclusions   were    drawn   eoncern- 

"^  If  the  manner  of  the  development  or  of  the  periodicity  of  a 
series  is  already  established,  we  may  try  to  measure  independently 
the  irregular  individual  fluctuations  occurring  in  the  series  super- 
imposed upon  the  regular  fluctuations,  and  to  compute  their  average. 
For  this  purpose  we  must  ascertain  the  value  for  every  year  which 
results  merely  from  the  general  development  or  periodicity,  and  in 
this  way  we  construct,  so  to  speak,  a  hypothetical  curve.  The 
deviation  from  the  hypothetical  curve  (i.  e.,  from  the  value  resulting 
in  the  hypothetical  curve  for  the  year  in  question)  of  the  actual 
series  must  then  be  ascertained  for  every  year  separately  and  the 
average  of  these  deviations  must  be  computed. 


DEFINITELY  LIMITED  MAGNITUDES  295 

ing  free-will,  which  naturally  caused  endless  controversy. 
As  a  matter  of  fact,  there  are  only  few  steady  time 
series  consisting  of  absolute  numbers.  Most  phenomena 
that  are  treated  statistically  are  closely  connected  with 
the  number  of  population.  But  the  latter  shows  a  more 
or  less  decided  variation  in  almost  all  countries.  Conse- 
quently, a  definite  trend  is  usually  expressed  in  the  various 
social  phenomena  whose  absolute  sizes  are  ascertained  statis- 
tically. Only  by  computing  relative  numbers  which  ex- 
press the  extent  of  the  phenomena  observed  per  thousand  of 
the  population  or  per  inhabitant  is  it  possible  to  obtain 
values  independent  of  the  variation  of  the  number  of  popu- 
lation. Therefore,  the  question  of  the  permanence  and  so- 
called  regularity  of  the  social  phenomena  can  only  be  solved 
by  investigating  the  fluctuations  shown  by  series  consisting 
of  relative  numbers.  These  series  belong  in  the  third 
group  and  their  dispersion  will  be  treated  in  the  following 
chapter. 

An  indubitable  connection  between  the  homogeneity  of 
the  series  and  its  dispersion  exists  in  the  time  series  formed 
of  absolute  numbers,  as  well  as  in  the  series  of  quantitative 
individual  observations.  Thus,  a  series  which  extends  over 
periods  in  which  the  phenomena  measured  were  exposed  to 
very  heterogeneous  influences,  undoubtedly  shows  greater 
variations  than  a  series  which  refers  to  a  period  that  in 
itself  is  homogeneous.  If  a  time  series  is  decomposed  into 
constituent  series  the  items  of  each  being  homogeneous  in 
character,  for  instance,  if  economically  favorable  and  un- 
favorable years  are  separated  when  investigating  the  fluctua- 
tions of  births  and  marriages,  then  constituent  series  are 
obtained  which  usually  show  much  smaller  fluctuations  than 
the  total  series.  Adolf  Wagner  divided  the  marriages 
in  Belgium  during  the  period  1841-1858  into  three  groups 
by  combining  years  of  the  same  industrial  character. 
He  distinguished  a  normal  period  consisting  of  the  years 
1841-1845,  an  unfavorable  period  consisting  of  the  years 


296     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

of  dearth,  1S46,  1847,  1854,  1855,  and  a  favorable  period 
consisting  of  the  years  of  low  prices,  1849,  1850,  1857,  1858. 
He  found  an  astonishing  regularity  for  each  period.®" 

••  Cf.  Gesetzmassigkeit  in  den  scheinbar  willkiirlichen  menschlichen 
Handlxmgen,  "  Comparative  Suicide  Statistics,"  p.  934. 


CHAPTER  IV 

THE  DISPERSION  OF  SERIES  OF  RELATIVE  Nn3IBERS 
AXD  AVERAGES  WHICH  CHARACTERIZE  MASSES 
LDIITED  IN  A  DEFINITE  WAY  (CONSTITUENTS  OF 
A  GREATER  TOTALITY)  IN  SOME  OTHER  RESPECT 
THAN  AS  REGARDS  THEIR  MAGNITUDES 

A.     THE      GENERAL      PROBLEM      (DISTrS'GLlSHES'G 
TBIE,  SPACE,  AND  QUALITATIVE  OR  QUANTITATIVE 

SERIES  J 

The  third  group  embraces  those  series  of  relative  num- 
bers and  averages  which  characterize  masses  limited  in 
a  definite  way  (constituents  of  a  greater  totality)  in  some 
other  respect  than  as  regards  their  magnitudes.  The  rela- 
tive number  computed  directly  for  the  greater  totality 
stands  as  a  mean  of  the  items  of  the  series,  if  these  are 
relative  numbers:  if  the  items  of  the  series  are  them- 
selves averages  of  quantitative  elements  of  observation 
(computed  for  the  constituent  masses),  then  the  average 
which  is  computed  for  the  totality,  and  which  is  logically 
analogous  to  the  items,  is  to  be  regarded  as  the  mean  of 
the  series. 

The  dispersion  of  a  time  series  consisting  of  relative 
numbers  or  averages  expresses  the  degree  of  stability  (stead- 
iness, constancy)  or  the  degree  of  variability  of  the  phe- 
nomenon measured.  Phenomena  which  nearly  coincide  for 
consecutive  periods  of  time,  or  which  give  rise  to  relative 
numbers  or  averages  but  slightly  different  from  the  general 
averages  of  a  longer  period,  are  "  stable  "  in  contradis- 
tinction to  the  *'  variable  "  phenomena  whose  values  pre- 
sent decided  time  fluctuations.     Such  time  series  may  be 

297 


298     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

classified,  therefore,  according  to  the  size  of  their 
measures  of  dispersion.  It  is,  however,  to  be  kept 
in  mind  that  time  series  of  relative  numbers  and 
averages,  like  time  series  of  absolute  numbers,®^  fre- 
quently exhibit  a  characteristic  conformation,  not  in 
the  grouping  of  the  items  about  the  mean  but  in 
some  other  way,  such  as  a  definite  evolution  or  a  definite 
periodicity.  This  characteristic  conformation  cannot  be 
established  by  merely  measuring  the  dispersion  of  the 
items  about  the  mean,  as  such  a  measure  takes  no  account 
of  the  order  of  the  items,  which  is  the  essential  element  in 
determining  the  evolutionary  or  periodic  character  of  the 
series.  Therefore,  in  this  connection,  chief  attention  will 
be  given  to  those  series  which  possess  no  marked  evolution 
or  periodicity,  while  the  investigation  of  the  series  which 
have  a  characteristic  conformation,  but  not  in  the  grouping 
of  the  items  about  the  mean,  will  be  considered  in  Appendix 

T  62 

Time  series  of  relative  numbers  and  averages  usually 
exhibit  relatively  slight  fluctuations  in  the  course  of  years. 
If  this  were  not  true,  then  most  statistical  data  would  pos- 
sess no  practical  value,  as  conclusions  could  not  be  based 
upon  them.  Small  time  fluctuations  are  to  be  found,  as  we 
should  expect,  primarily  in  the  fields  of  anthropology  and 
meteorology.  The  average  height  and  weight  of  the  inhab- 
itants of  a  country  change  very  little.  Likewise,  there  is 
little  variation  in  the  average  temperature,  barometric 
height,  or  rainfall  of  a  given  country.  Also  in  the  fields 
of  social,  moral,  and  economic  statistics  there  exists  rela- 
tively great  stability,  as  is  evidenced  by  the  sex-ratio  among 
births,  the  average   and  the  normal  lengths  of  life,  the 

'*  Compare  with  p.  294  f,,  above. 

"It  is  self-evident  that  the  dispersion  of  evolutionary  or  periodic 
series  can  also  be  measured.  Such  measurements,  however,  are  not 
concerned  with  the  essential  nature  of  the  series  and  possess  sig- 
nificance only  in  individual  cases  for  some  particular  purpose. 


RELATIVE  NUMBERS  AND  AVERAGES  299 

birth,  marriage,  and  death  rates,  the  percentage  of  crimes, 
the  consumption  of  food  per  capita,  the  average  income, 
the  average  wage,  and  numerous  other  values  which  exhibit 
very  slight  fluctuations  during  consecutive  years/'^^ 

The  attitude  of  statisticians  in  regard  to  the  significance 
of  the  stability  of  statistical  values  has  changed  decidedly 
during  the  last  few  decades.  When  the  relative  stability 
of  a  large  number  of  social  phenomena  was  first  established 
it  was  believed  that  an  extremely  remarkable  discovery  had 
been  made.  The  regularity  of  births  and  deaths  spelled 
"  gottliche  Ordnung  "  to  Siissmilch,  while  the  invariability 
(Gesetzmassigkeit)  of  demographic  and  moral  statistical 
phenomena  filled  Quetelet  and  his  followers  with  admira- 
tion for  the  ''  natural  law  "  that  seemed  revealed.  The 
constancy  observed  for  a  few  phenomena,  limited  periods 
of  time,  and  a  restricted  area  was  represented  as  being 
universally  true  and,  where  observations  were  wanting, 
constancy  was  presumed  without  further  investigation.  The 
many  data  collected  in  the  course  of  time  have  since  sho^Mi, 
however,  that  a  variety  of  degrees  of  stability  (or  variabil- 
ity) exist,  and  that  the  degree  of  variability  of  a  single 
phenomenon  changes  in  the  course  of  time  and  from  country 
to  country.  The  stability  of  social  (demographic,  moral 
and  economic)  phenomena  is,  therefore,  at  present  not 
considered  a  general  law,  and  where  constancy  is  found  it 
is  explained  without  the  aid  of  metaphysics  and,  as  a  rule, 
without  postulating  natural  law.  The  constancy  or  varia- 
bility of  a  social  phenomenon  is,  according  to  present  opin- 
ion, due  to  the  constancy  or  variability  of  the  conditions 
upon  which  the  phenomenon  in  question  is  dependent.  If 
the  complex  of  causes  which  produces  the  phenomenon  re- 

•*aThe  author  is  in  error  in  stating  that  the  variation  in  the 
average  temperature,  rainfall,  and  barometric  height  of  a  given 
country  is  small,  as  it  varies  by  as  much  as  100^  in  certain  dis- 
tricts. Likewise,  certain  series  of  social  statistics,  such  as  annual 
divorce  rates,  have  wide  fluctuations. — Tbanslatob. 


300     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

mains,  in  the  main,  unchanged,  then  there  is  no  possibility 
for  an  essential  change  in  the  phenomenon.  If,  however, 
any  of  the  causes  is  altered  then  the  dependent  phenomenon 
must  change. 

The  phenomena  with  which  statistics  is  concerned  depend 
both  upon  natural  and  upon  social  causes.  The  former 
predominate,  of  course,  in  the  fields  of  meteorology  and 
anthropology.  In  demography  the  limitations  of  human 
life  and  of  the  human  reproductve  period  are  conditions 
prescribed  by  nature.  The  ratio  of  the  sexes  also  appears 
to  be  ruled  by  natural  law.  Probably  it  is  also  to  be  con- 
sidered a  result  of  natural  law  that  a  great  part  of  the 
children  born  are  lacking  in  vitality  and  succumb  to  a 
special  mortality.^^  The  general  composition  of  a  population 
according  to  sex  and  age  is,  therefore,  determined  by  natu- 
ral law,  and  it  can  alter  but  slowly.  The  natural  stability 
of  the  demographic  divisions  of  population  necessarily  re- 
sults in  relatively  great  social  and  economic  stability  and 
hence  in  a  certain  regularity  of  social  and  economic  mass- 
phenomena.^* 

Various  social  causes  are  acting  in  the  general  frame- 
work of  society.  These  causes,  such  as  occupation,  economic 
condition,  education,  etc.,  exercise,  as  is  their  nature 
and  intensity,  a  varying  degree  of  influence  upon 
demographic,  moral,  and  economic  phenomena.  These 
various  degrees  of  influence  are,  however,  quite  compatible 
with  the  constancy  of  the  phenomena  named.  One  can 
start  with  the  idea  of  dividing  the  population  according  to 
occupation,  economic  condition,  education,  etc.,  into  more 
homogeneous  groups  which  participate  to  a  varying  degree 
in  demographic,  moral  and  economic  phenomena.     The  eco- 

""  Lexis,  Abhandlungen  zur  Theorie  der  Bevolkerungs-  und  Moral- 
statistik,  V,  "  Concerning  the  Causes  of  the  Slight  Variability  of 
Statistical  E,elative  Numbers,"  p.  87. 

"*  Ibid.  pp.  353  and  94.  Also  see  the  articles  "Gesetz"  and 
"  Moralstatistik  "  by  Lexis  in  the  Handw.  der  Staatsw. 


RELATIVE  NUMBERS  AND  AVERAGES  301 

nomic  and  intellectual  conditions  of  a  definite  group  of  the 
population,  and  the  hygienic  conditions  under  which  the 
members  of  that  group  live,  naturally  cause  a  definite 
mortality;  they  make  it  possible  for  a  definite  percentage 
of  the  young  people  to  marry  and  to  have  children;  they 
also  expose  the  members  of  the  group  to  definite  tempta- 
tions to  commit  crimes  of  one  kind  or  another  and  thus 
cause  a  definite  rate  of  criminality.  The  various  groups 
of  the  population  have  also,  of  course,  various  economic 
requirements  and  produce  various  economic  goods.  As 
long  as  the  conditions  of  life  of  the  members  of  the  different 
groups  do  not  change,  so  long  will  each  group  develop  the 
same  intensity  of  various  demographic,  economic,  and  moral 
phenomena.  If,  at  the  same  time,  the  whole  population 
continues  to  be  composed  of  the  same  constituent  homo- 
geneous groups,  then,  of  course,  the  whole  population  will 
continuously  exhibit  demographic,  moral,  and  economic  rel- 
ative numbers  and  averages  of  the  same  size.  This  is  true 
— always  assuming  that  the  composition  of  the  population 
remains  the  same — even  if  the  number  of  population  in- 
creases or  diminishes,  since  the  sizes  of  both  relative  num- 
bers and  averages  do  not  necessarily  vary  with  the  number 
of  observations.  The  death  rate  (number  of  deaths  per  thou- 
sand living)  or  the  per  capita  consumption  of  meat  may 
have  the  same  values  for  wholly  different  numbers  of  popu- 
lation. Of  course,  if  the  composition  of  the  population 
changes  in  any  manner,  that  is,  if  the  proportion  existing 
among  the  various  constituent  homogeneous  groups  changes, 
then  an  effect  will  always  be  noticed  in  some  one  or  other 
of  the  phenomena  comprehended  by  statistics.  Thus,  for 
example,  an  increase  of  industrial  activity,  which  reduces 
the  numbers  of  the  unemployed  and  places  many  people  in 
a  better  economic  position,  will  diminish  the  mortality"  and 
the  number  of  crimes  against  property,  but  will  increase 
the  number  of  marriages ;  an  economic  crisis,  on  the  other 
hand,  swells  the  weakest  economic  classes,  and  these  classes 


302     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

will  contribute  more  strongly,  according  to  their  peculiari- 
ties of  conduct,  to  the  various  demographic  and  moral 
phenomena. 

From  what  has  been  said  it  follows  that  general  laws  of 
stability  in  the  province  of  social  statistics  cannot  be  estab- 
lished. The  relatively  great  stability  of  social  phenomena 
can  be  explained  by  the  permanence  of  the  underlying 
cause-complexes ;  that  is,  the  stability  naturally  results  since 
the  economic  institutions  and  mental  make-up  of  a  given 
population  change  very  gradually,  while  statistical  series 
usually  refer  to  relatively  short  periods  of  time.  This  sta- 
bility has,  however,  nothing  absolute  in  it,  but  it  is  merely 
an  empirical  fact  of  observation  arising  from  definite  con- 
crete conditions  and  it  may  at  any  time  give  way  either 
to  marked  fluctuations  or  enter  a  definite  evolution  follow- 
ing some  political,  economic,  technical,  or  other  cause.  If 
we  desire  to  use  the  term  ''  statistical  law  "  in  its  broad 
sense  in  referring  to  the  stable  phenomena  that  we  have 
been  considering,  we  must  keep  in  mind  that  we  are  dealing 
with  purely  empirical  social  laws,  which  are  to  be  con- 
sidered merely  as  historical  categories.^^ 

In  statistical  literature  attempts  have  been  made  to  divide 
time  series  into  groups  of  a  definite  character,  which  groups 
would  also  exhibit  various  degrees  of  stability.  These 
attempts  have,  as  will  be  shown,  led  to  no  satisfactory 
classification,  as  essentially  related  series  often  possess  quite 

"  According  to  Wundt  it  is  not  correct  to  speak  of  an  empirical 
"  law "  with  reference  to  the  time-constancy  of  a  phenomenon,  as 
there  is,  in  this  case,  no  pertinent  characteristic  indicating  a  causal 
relationship.  On  the  contrary,  those  regularities  are  to  be  designated 
as  empirical  laws  in  which  the  mass-phenomena  stand  in  a  func- 
tional relation  to  definite  time-  or  space- values  (for  instance,  a 
regular  evolutionary  tendency),  or  in  which  there  is  directly  con- 
cerned a  causal  relation  between  mass-phenomena  independent  of 
one  another  (for  instance,  in  the  proof  of  the  various  death  rates 
in  different  occupations).  (See  Logik,  Vol.  II,  Pt.  II,  Logik  der 
Geisteswissenschaften  (1895),  pp.  144,  464,  472  f.) 


I 


RELATIVE  NUMBERS  AND  AVERAGES  303 

different  dispersion,  while,  on  the  other  hand,  the  dispersion 
of  quite  unlike  series  is,  many  times,  the  same. 

Thus  the  theory  has  been  advanced  that  statistical  phe- 
nomena exhibit  varying  degrees  of  stability  according  to  the 
predominance  of  natural  events  or  of  acts  of  the  human  will 
in  causing  them.  It  is  very  significant  that  opposite  views 
are  held  by  eminent  writers;  some  holding,  on  the  one 
hand,  that  phenomena  that  depend  upon  natural  factors 
are  more  stable  and  others  holding,  on  the  other  hand, 
that  it  is  the  actions  controlled  by  the  will,  such  as  mar- 
riage, criminality,  and  suicide,  that  recur  with  greater 
regularity. 

In  general,  it  is  certainly  true  that  natural  causes  bring 
about  fluctuations  that  are  often  between  narrow  limits. 
The  sex-ratio  of  births,  which  seems  to  be  determined  ex- 
clusively by  natural  causes,  is  the  most  stable  demographic 
phenomenon  which  we  know.  Likewise,  the  average  stature 
of  the  population  of  any  country  scarcely  varies,  although, 
it  is  true,  this  value  is  also  influenced  by  social  causes 
which  help  or  hinder  growth.  On  the  other  hand,  other 
phenomena,  also  chiefly  dependent  upon  natural  causes, 
show  greater  fluctuations.  For  instance,  meteorological 
phenomena  (average  rainfall,  average  temperature,  average 
barometric  height,  etc.)  do  not  only  vary  widely  with 
the  seasons  but  also  from  year  to  year.  Consequently, 
agricultural  products,  which  are  dependent  upon  meteoro- 
logical conditions,  likewise  show  considerable  fluctuations. 
It  is  quite  different,  however,  with  certain  phenomena  which 
are  dependent  chiefly  upon  the  human  will,  such  as  mar- 
riages, crimes  and  suicides,  which  show  a  remarkable  regu- 
larity during  the  course  of  years.  The  comparison  of  these 
"  voluntary  "  phenomena  with  mortality,  which  is  subject 
to  natural  laws,  is  the  chief  origin  of  the  thesis  of  the 
greater  regularity  of  voluntary  events.  It  is  true  that  there 
are  greater  fluctuations  in  mortality  rates  than  in  marriage. 


304     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

crime,  or  suicide  rates.  But  the  death  rate  is  a  phenomenon 
by  no  means  exclusively  determined  by  natural  causes.  The 
extreme  limit  and,  perhaps,  the  normal  length  of  men's 
lives  are  fixed  by  natural  law.  The  normal  lifetime,  as  a 
matter  of  fact,  is  extraordinarily  stable.  The  frequency  of 
deaths,  however,  which  does  not  arise  entirely  from  the 
normal  mortality  but  largely  depends  upon  the  mortality 
of  children,  is  in  a  great  measure  dependent  upon  economic 
relations  and  fluctuates  with  each  economic  change.  A 
business  depression  throws  a  large  number  of  men  out  of 
work  or  at  least  forces  them  to  adopt  modes  of  life 
much  inferior  and,  therefore,  dangerous  to  health;  an  im- 
provement of  the  economic  situation  also  influences  mor- 
tality by  enabling  many  people  to  adopt  a  mode  of  life 
more  advantageous  to  health.  The  number  of  marriages, 
as  well  as  crimes  and  suicides,  is,  to  be  sure,  also  influenced 
by  economic  conditions,  but,  in  general,  to  a  less  degree 
than  is  mortality.  Marriages,  crimes,  and  suicides  are 
events  which  are  not — like  deaths — independent  of  the 
human  will,  but  they  are  conscious  and,  as  a  rule,  maturely 
considered  human  acts.  The  motives  which  lead  to  mar- 
riages, crime,  or  suicide  are  not  exclusively  economic;  that 
is  to  say,  these  phenomena  are,  to  a  great  extent,  caused  by 
motives  which  are  independent  of  the  fluctuations  of  eco- 
nomic activity.  In  consequence  of  the  relative  permanence 
of  the  general  social  relations  and  of  the  mental  constitution 
of  the  population,  the  non-economic  motives  act  with  nearly 
the  same  intensity  during  a  course  of  years  and  sweep 
nearly  the  same  proportions  of  the  population  to  marriage, 
crime,  and  suicide  year  in  and  year  out — of  course  with 
the  difference  that  each  year  other  individuals  are  vehicles 
of  the  motives — and  thus  a  surprising  regularity  of  "  vol- 
untary "  action  results.  In  this  sense  it  is  undoubtedly 
certain  that  human  will  is,  in  general,  an  element  of  stabil- 
ity.   A  general  proposition  that  "  voluntary  "  actions  are 


RELATIVE  NUMBERS  AND  AVERAGES  305 

always   more   stable   than   occurrences   in   which   natural 
causes  predominate  cannot,  however,  be  established.^® 

The  fact  of  the  relatively  great  stability  of  crimes, 
suicides,  and  similar  moral-statistical  phenomena  has  led  to 
many  philosophical  controversies  concerning  the  freedom 
of  the  will.  A  number  of  the  earlier  statistical  writers 
interpreted  the  regularity  of  moral-statistical  phenomena 
as  "  natural  law  "  and  believed  the  freedom  of  the  will 
to  be  denied  by  this  regularity,  or  at  least  to  be  confined 
between  definite  limits.  Quetelet  initiated  this  line  of 
argument.  He  put  forth  the  idea  of  natural  law  for  mass- 
phenomena,  without,  however,  excluding  individual  freedom 
of  will,®^  This  manifestly  unsatisfactory  formulation  was 
accepted  by  numerous  statisticians,  especially  by  the  Italian 
statisticians  Messedaglia,  Corradi,  Bodio,  and  Morpurgo, 
without  solution  of  its  inherent  contradiction.*^^  Other 
writers,  influenced  by  Quetelet,  especially  Buckle,  went 
even  further  than  he  and  used  statistical  regularity  as  an 
argument  for  a  thoroughgoing  determinism.  Strong  oppo- 
sition to  this  development  arose  among  the  champions  of 
the  freedom  of  the  will,  especially  among  the  Germans 
Wappaus,  Riimelin,  Rheinisch,  and  others,  who  would  admit 
no  subordination  of  individual  action  to  the  "  laws  "  ruling 

'*  This  conclusion  also  results  from  consideration  of  statistical 
series  from  the  standpoint  of  the  theory  of  probability.  Such  con- 
sideration shows  that  the  laws  of  chance  are  independent  of  the 
causes  which  are  operative  in  individual  cases.  (See  the  review 
of  Lexis'  Abhandlungen  zur  Theorie  der  Bevolkerungs-  und  Moral- 
statistik  by  v.  Bortkiewicz  in  the  Jahrb.  f.  Nat.  und  Stat.,  3rd 
series,  Vol.  XXVII,  1904,  p.  253.) 

*^  Quetelet  held,  moreover,  that  this  moral-statistical  regularity 
was  not  entirely  invariable.  He  recognized,  for  instance,  that  the 
progress  of  civilization  must  carry  with  it  a  decrease  of  mortality 
and  of  criminality.  But  he  believed  in  the  possibility  of  very  gradual 
changes  only.  (See  Uber  den  Menschen,  German  edition,  1838,  pp. 
10  f.,  557  f.) 

"  See  Meitzen,  Geschichte,  Theorie  und  Technik  der  Statistik,  2nd 
ed.,  p.  61. 


306     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

masses,  but  they  offered  no  satisfactory  explanation  of  the 
consistency  of  freedom  of  the  will  and  social  regularity. 
Adolf  Wagner  called  attention  in  1864  to  the  contra- 
diction between  the  regularity  of  mass-phenomena,  which 
he  demonstrated  statistically,  and  the  freedom  of  the  will, 
to  which  he  declared  his  adherence,  and  said  that 
the  removal  of  this  contradiction  must  be  the  purpose  of 
further  scientific  research.*^®*  It  was  Drobisch's  reduction 
of  the  acts  of  individuals  to  the  volition  resulting  from 
definite  motives  that  first  gave  the  correct  standpoint  for 
explaining  the  relation  between  the  will  of  the  individual 
and  the  actions  of  the  many. 

The  modern  explanation  based  upon  the  writings  of 
Drobisch,  Schmoller,  and  Knapp,  and  formulated  most  pre- 
cisely by  Lexis,  ascribes  the  regularity  of  moral-statistical 
phenomena  to  the  relative  permanence  of  the  social  con- 
ditions which  are  the  fundamental  causes  of  such  phenom- 
ena, the  permanency  of  these  conditions  regularly  giving 
rise  to  motives  leading  to  the  same  acts.  This  explanation 
is  independent  of  the  metaphysical  question  of  the  existence 
or  non-existence  of  free-will,  and  it  is,  therefore,  the  gen- 
eral conviction  that  statistics  has  not  to  solve  and  cannot 
solve  this  question.®^** 

The  extensive  moral-statistical  material  now  available 
shows  also  that  the  regularity  assumed  by  the  earlier 
statisticians  is  usually  not  so  great  as  the  analogy  with 
natural  law  would  make  necessary.  The  mathematical 
statisticians  in  particular  have  shown  that  even  the  most 
stable  series  of  moral  statistics  which  have  been  observed 
are  affected  by  fluctuations  that  are  at  least  as  great,  and 

"a  Gesetzmassigkeit  in  den  scheinbar  willkiirlichen  menschlichen 
Handlungen,  p.  79. 

"bMr.  F.  H.  Hankins  has  clearly  explained  the  attitude  of 
modern  statisticians  toward  this  question  in  his  Adolphe  Quetelet 
as  Statistician  (Columbia  University  Studies  in  History,  etc.,  Vol. 
XXXI,  No.  4). — Translatob. 


RELATIVE  NUMBERS  AND  AVERAGES  307 

usually  mueli  greater,  than  the  accidental  errors  of  the 
values  obtained  empirically  in  games  of  chance.  From  this 
it  follows  that  even  in  such  stable  series  there  are  disturb- 
ing factors  whose  effects  are  as  great  as  those  of  accidental 
causes.  Free-will  can  enter  as  one  of  these  disturbing  fac- 
tors operating  in  a  manner  analogous  to  accidental  causes. 
The  actual  fluctuations  of  series  of  moral  statistics  offer, 
therefore,  sufficient  scope  for  the  effects  of  free-will,  under- 
stood in  the  sense  of  an  accidental  cause.®^ 

The  second  criterion  that  writers  have  believed  would 
divide  time  series — and,  in  this  case,  merely  time  series 
consisting  of  relative  numbers — into  two  groups  of  different 
degrees  of  stability  is  the  differentiation  of  such  series  into, 
first,  series  of  relative  numbers  which  represent  the  com- 
position of  definite  masses  and,  second,  series  of  relative 
numbers  which  give  the  frequency  or  intensity  of  definite 
phenomena.  Relative  numbers  of  the  first  kind  are,  as  a 
rule,  subordinate  numbers  which  bear  the  character  of 
relative  (analytical,  secondary)  probability,  as,  for  example, 
the  percentage  of  men  or  women  in  a  whole  population. 
Relative  numbers  of  this  kind,  however,  may  sometimes  be 
coordinate  numbers — as  the  number  of  women  per  one  hun- 
dred men — which  correspond  to  a  known  function  of  a 
relative  probability.  Relative  numbers  of  the  second  kind, 
which  give  the  frequency  or  intensity  of  a  phenomenon,  are 
always  coordinate  numbers  which  can  bear  the  character 
of  a  "  genetic  "  probability.  In  this  class  belong  the  mor- 
tality rate  and  the  probability  of  death,  the  birth  rate,  the 
marriage  rate,  etc. 

It  has  been  observed  that  series  of  relative  numbers  which 
represent  the  composition  of  definite  masses  exhibit  a  much 
more  remarkable  stability  in  time,  that  is,  the  dispersion  is 
less,  than  do  the  coordinate  numbers  which  show  the  fre- 

°°  Cf.  Tschuprow,  "  Die  Aufgaben  der  Theorie  der  Statistik," 
Sehmoller's  Jahrbuch,  29  Jahrgang  (1905),  p.  461  f.,  and  Wester- 
gaard,  Die  Grundzuge  der  Theorie  der  Statistik,  p.  282. 


308     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

quency  or  intensity  for  the  same  space  of  time  of  the 
phenomenon  whose  inner  composition  is  represented  by 
the  former  series.  Most  striking  are  the  different  degrees 
of  stability  shown  by  a  comparison  of  the  birth  rates 
(frequency  or  intensity)  and  the  sex-ratios  of  births  (inner 
composition).  The  birth  rates  may  show  very  considerable 
fluctuations  while  the  sex-ratios  remain  constant.  This 
fact,  however,  evidently  depends  upon  the  further  fact  that 
the  sex  composition  of  births  is  exclusively  or  almost  ex- 
clusively determined  by  natural  law,  while  birth  rates  de- 
pend upon  various  fluctuating  social  factors.  Also  the  fact 
that  the  percentage  of  stillborn  among  all  births  scarcely 
changes,  in  spite  of  the  fluctuations  of  the  birth  rates,  is 
probably  most  intimately  related  to  the  natural  causes 
which  bring  about  still-births.  But  we  also  discover  similar 
conditions  in  phenomena  whose  structure  is  in  no  way 
primarily  determined  by  natural  causes.  For  example,  if 
those  marrying  be  classified  according  to  age  and  conjugal 
condition  the  classes  will,  as  a  rule,  be  much  more  stable 
than  the  general  marriage  rate.  Likewise,  the  classes  of 
criminals  according  to  age  and  sex  fluctuate  much  less 
than  general  criminality,  and  suicides  classified  according 
to  age,  sex,  and  the  particular  form  of  death  are  less 
variable  than  the  total  number  of  suicides.  The  number 
of  emigrants  per  thousand  of  population  varies  extraor- 
dinarily, nevertheless  the  age  and  sex  composition  remains 
about  the  same  from  year  to  year.  The  regularity  in  the 
proportion  of  various  classes  of  mail,  in  spite  of  the  increase 
in  the  volume  of  mail  matter,  is  well  known.  The  per- 
centage of  letters  registered,  of  letters  free  of  postage, 
of  postcards  fluctuates  very  little;  likewise  the  percentage 
of  dead  letters  remains  almost  constant. 

This  great  constancy  of  certain  subordinate  numbers  is, 
however,  not  difficult  to  explain.  The  explanation  is  that 
the  factors  which  cause  the  variations  in  frequency  (in- 
tensity)  of  the  respective  phenomena  exercise  no,  or  but 


1 


RELATIVE  NUMBERS  AND  AVERAGES  309 

slight,  influence  on  the  distribution  of  the  events  into  the 
various  subdivisions  considered.  Thus,  economic  factors 
affect  the  number  of  marriages  and  cause  the  marriage  rate 
to  fluctuate.  But  these  economic  factors  (economic  expan- 
sion, crises,  etc.)  operate  more  or  less  uniformly  on  all 
marriageable  persons  regardless  of  their  social  position 
or  age,  and  the  subdivisions  of  the  people  marrying  accord- 
ing to  social  position  or  age  remain  relatively  the  same, 
regardless  of  the  changes  in  the  frequency  of  marriage. 
The  percentage  of  registered  letters  expresses  the  view  of 
the  public  as  to  the  reliability  of  the  postal  service  and  the 
risk  of  sending  an  ordinary  letter.  If  this  view  remains 
the  same  the  percentage  of  registered  letters  will  remain 
constant  even  though  the  total  number  of  letters  is  greater. 
No  generalization  can  be  drawna,  however,  concerning 
this  group  of  eases,  in  which  the  inner  composition  is 
probably  more  stable  than  the  frequency  or  intensity  of 
the  phenomenon.  The  question  whether  the  frequency  of 
a  phenomenon  fluctuates  more  than  do  the  classes  obtained 
by  dividing  the  individual  items  into  definite  categories, 
must  be  solved  de  novo  for  each  phenomenon.  Finally,  it 
also  happens  that  where  the  classes  exhibit  extraordinary 
stability,  changes  appear  which  must  not  be  overlooked. 
Thus,  at  present  a  tendency  is  shown  in  most  countries, 
in  consequence  of  the  decrease  of  mortality  and  the  devel- 
opment of  industry,  for  the  number  of  first  marriages  of 
both  sexes  to  increase  at  the  expense  of  second  or  later 
marriages.  In  the  statistics  of  suicides,  classified  according 
to  the  various  methods  of  death,  absolute  constancy  is  not 
possible  on  account  of  the  changing  classification;  our 
modern  life  gives  rise  to  new  and  formerly  unkno\vn 
methods,  such  as  death  from  a  moving  train.  The  classes 
of  deaths  according  to  cause  in  general  show  a  remarkable 
constancy.  In  spite  of  this,  certain  causes  of  death  fluctu- 
ate considerably  or  show  a  definite  tendency  of  develop- 
ment.    Thus,    in    England    during   the   period    1871-1890 


310     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

there  was  a  decrease  of  mortality  from  consumption,  small- 
pox, and  typhoid  and  related  fevers,  while  the  mortality 
from  cancer  greatly  increased.^**  Statistical  masses  in 
which  the  inner  composition  changes  more  in  the  course 
of  time  than  the  intensity  are,  consequently,  not  unknown. 
Thus  Ottingen  has  given  statistical  data  concerning  the 
literary  productivity  of  Gemany,'^"*  from  which  it  appears 
that  during  the  years  1850-1875,  when  the  annual  produc- 
tion was  tolerably  constant,  the  relative  proportions  of 
single  classes  of  writing  altered  considerably.  There  was 
an  increase  of  technical  books  relating  to  industry  and  a 
decrease,  especially,  of  theological  and  religious  literature, 
whose  place  was  taken  by  pedagogical  books  of  a  popular 
or  juvenile  character. 

For  series  consisting  of  relative  numbers  or  averages, 
just  as  for  those  consisting  of  absolute  numbers,  there  is  a 
connection  between  the  homogeneity  of  the  series  and  its 
dispersion.  If  a  time  series  covers  a  homogeneous  period 
it  naturally  exhibits  smaller  fluctuations  than  one  covering 
a  period  during  which  the  factors  chiefly  influencing  the 
phenomenon  represented  have  varied  in  strength.  If  one 
separates  the  economically  good  and  bad  periods  from 
each  other,  then  less  dispersion  is  shown  by  those  series 
dependent  upon  economic  conditions  (mortality,  crimes, 
suicides,  etc.)  than  by  combining  years  of  business  depres- 
sion and  expansion. 

Aside  from  economic  conditions  other  causal  factors  may 
also  be  concerned.  Thus,  the  meteorological  conditions  of 
a  year  doubtless  exercise  a  great  influence  upon  the  mor- 
tality of  that  year.  In  general,  we  find  that  years  with 
hot  summers  and  cold  winters  have  a  high  mortality  and, 
inversely,  that  a  cool  summer  and  mild  winter  are  health- 
ful. If  one  should  select  from  a  period  of  years  those 
which  have  approximately  the  same  conditions  as  regards 

'"'  Compare  G.  v.  Mayr,  Bevolkerungsstatistik,  p.  323. 
»»a.MoraIstatistik,  p.  555  f. 


RELATIVE  NUMBERS  AND  AVERAGES  311 

temperature,  and  should  study  the  fluctuations  of  the 
mortality  during  these  years,  one  would  find  a  narrower 
range  of  deviations  from  the  average  than  is  shown  by  all 
the  years  of  the  period  regardless  of  temperature.'^^ 

We  now  leave  time  series  to  consider  geographical  series 
of  relative  numbers  or  averages.  The  dispersion  of  such 
series  gives  a  standard  for  measuring  the  variability  of 
phenomena  with  reference  to  space.  There  may  be  values 
for  various  countries  or  their  subdivisions  which  differ  very 
little  from  each  other  and  from  their  average ;  on  the  other 
hand,  values  for  different  geographical  divisions  may  differ 
widely.  In  general,  geographical  series,  as  long  as  the 
items  do  not  refer  to  divisions  of  a  single  homogeneous 
country,  exhibit  greater  fluctuations  than  do  time  series. 
This  is  very  easily  understood.  That  time  series  made 
up  of  demographic,  moral,  or  economic  statistics  exhibit 
but  small  fluctuations  rests  upon  the  fact  that,  as  a  rule, 
only  a  few  decades  are  considered,  during  which  social 
conditions  usually  change  very  little.  On  the  contrary, 
it  is  to  be  expected  that  values  drawn  from  different 
countries  will  usually  show  greater  differences  for  different 
countries  develop  along  a  great  variety  of  lines  because  of 
differences  of  geographical  position,  climate,  or  other 
natural  attributes.  In  addition  there  are  often  ethnological 
differences  of  population,  and  also  differences  of  political, 
economic,  and  psychological  development,  which  are  the 
results,  not  of  tens  of  years,  but  of  hundreds  of  years  of 
history.  For  these  reasons  there  are  essential  differences 
of  social  and  demographic  conditions  among  the  popula- 
tions of  different  lands,  which  are  reflected  to  a  greater 
or  less  degree  in  all  statistical  phenomena.  They  appear, 
however,  to  the  smallest  degree  in  those  phenomena  which 
seem  to  rest  upon  definite  natural  laws  rather  than  upon 
social  factors.  Thus,  there  is  an  extraordinary  degree  of 
correspondence  among  the  sex-ratios  at  birth  in  various 

'*  Cf.  Westergaard,  Die  Grundziige,  etc.,  p.  19. 


312     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

countries.'^^  Greater  differences  are  to  be  found  in  the 
mean  and  normal  length  of  life  of  various  populations. 
That  meteorological  and  anthropometric  averages  (rainfall, 
temperature,  barometric  height,  stature  and  weight  of  the 
inhabitants,  etc.)  vary  decidedly  for  different  countries  is 
well  known.  Likewise,  the  differences  of  most  demographic 
and  economic  relative  numbers  and  averages  are  con- 
siderable, but  not  such  as  to  enable  us  to  make  any  general 
propositions  concerning  the  dispersion  of  geographic  series. 
Various  phenomena  (birth  rates,  mortality,  the  per  capita 
meat  consumption,  etc.)  as  a  rule  show  various  degrees  of 
dispersion,  and  it  is,  therefore,  necessary  to  depend  upon 
actual  observation  to  determine  the  kind  and  magnitude 
of  the  geographical  differences  of  each  phenomenon. 

Because  of  the  fundamental  differences  which,  as  a  rule, 
exist  between  various  countries  as  regards  the  composition 
and  activity  of  their  populations,  we  usually  do  not 
compute  general  averages  for  several  countries  together, 
hut  limit  ourselves  to  stating  the  minimum  and  maximum 
which  express  the  range  within  which  the  values  for  all 
countries  appear.  The  computation  of  an  average  em- 
bracing all  of  the  countries  in  the  series  and  the  use  of 
it  as  a  basis  of  comparison  appears  to  be  allowable  only 
when  the  various  countries  are  more  or  less  similar  in 
nature ;  for  example,  countries  which  possess  the  same  type 

^*  In  almost  all  countries  (according  to  Bodio's  compilation  based 
for  the  greater  part  upon  the  years  1887-1901)  there  are  104-106 
boys  to  100  girls  among  living  births.  England,  with  a  ratio  of 
103.4,  is  the  only  country  under  104,  while  only  Spain,  Portugal, 
Greece,  Roumania,  and  Connecticut  have  over  106.  It  is  possible 
that  in  a  geographical  comparison  the  differences  in  the  ratio  are  due 
to  racial  differences.  Edgeworth  has  ascribed  the  exceptionally  large 
excess  of  boys  in  Wales  to  the  Celtic  descent  of  the  inhabitants. 
(Journ.  of  the  Roy.  Stat.  Soc,  1898,  p.  130  f. )  It  is  known  that  there 
is  a  greater  excess  of  boys  among  the  Jews,  which  may  depend, 
however,  according  to  G.  v.  Mayr  ( Bevolkerungsstatistik,  p.  188), 
upon  the  fact  that  crossing  is  unusual  and  inbreeding  more  common. 


RELATIVE  NUMBERS  AND  AVERAGES  313 

of  civilization,  as  those  of  middle  Europe,  or  where  the 
items  of  these  series  do  not  relate  to  different  countries  but 
to  parts  of  a  single  country,  within  which  we  may  expect 
approximate  coincidence.  Thus,  it  is  customary  to  com- 
pare the  density  of  population  of  sections  of  a  nation 
with  the  national  average,  that  is,  to  ascertain  in  what 
manner  the  relative  numbers  which  express  the  density  of 
a  district  are  distributed  about  the  relative  number  (found 
by  dividing  the  total  population  by  the  total  area)  which 
expresses  the  density  of  population  of  the  entire  nation. 
In  the  same  way,  it  can  be  found  out  in  what  manner  the 
average  income  of  the  residents,  the  average  age  of  those 
marrying,  etc.,  of  the  several  districts  are  grouped  about 
corresponding  national  averages. 

The  proposition  concerning  the  connection  of  the  homo- 
geneity of  series  and  their  dispersion  holds  for  geographical 
series  of  relative  numbers  or  averages  as  well  as  for  the 
corresponding  time  series.  That  is,  less  dispersion  is  ob- 
tained by  decomposing  such  series  into  homogeneous  sub- 
divisions. For  example,  if  one  differentiates  nations  as 
healthful  or  unhealthful  according  to  their  various  climatic 
or  meteorological  characteristics,  then,  in  all  probability, 
the  series,  such  as  death  rates,  relating  to  each  homogeneous 
group  of  nations  will  show  much  less  dispersion,  that  is, 
less  deviation  from  the  average,  than  the  series  relating 
to  all  nations  regardless  of  climatic  or  meteorological  char- 
acteristics. 

Finally,  quantitative  and  qualitative  series  of  relative 
numbers  or  averages  will  be  discussed  with  reference  to 
their  dispersion,  in  order  to  get  a  picture  of  the  grouping 
of  the  items  about  their  average.  We  compute  the  death 
rates  in  various  occupations  and  determine  between  what 
limits  and  in  what  manner  the  various  death  rates  group 
themselves  about  the  average  death  rate  of  the  whole 
population.  Likewise,  a  series  may  be  composed  of  the 
average   wages   of  different   occupations   and   the   group- 


314     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

ing  of  these  about  the  general  average  may  be  deter- 
mined. 

But  there  are  reasons,  peculiar  to  the  nature  of  qualita- 
tive and  quantitative  series,  that  make  the  investigation 
of  the  dispersion  of  such  series  of  less  significance  than 
that  of  geographical  series  and  time  series.  If  a  phenom- 
enon (such  as  mortality)  remains  constant  or  fluctuates 
but  little  during  a  term  of  years,  we  could  not  have  pre- 
dicted it  as  a  matter  of  course.  It  is  only  by  study  of  the 
actual  figures  that  the  dispersion  is  disclosed  and  a  measure 
of  the  variability  of  the  phenomenon  with  reference  to  time 
is  obtained.  Likewise,  it  cannot  be  predicted  whether  a 
definite  phenomenon  will  appear  uniformly  or  with  varying 
intensity  in  different  geographical  districts ;  investigation 
of  the  dispersion  of  the  geographical  series  in  question  must 
give  the  information. 

It  is  quite  different  with  qualitative  and  quantitative 
series.  Such  series,  as  a  rule,  are  built  upon  the  basis 
of  some  mark  of  differentiation  which  we  already  know 
or,  at  least,  assume  to  possess  causal  significance  for  the 
phenomenon  represented  by  the  series.  Thus,  we  divide 
those  dying  according  to  occupation  or  economic  condition, 
because  we  know  that  these  influence  mortality  and  that 
the  members  of  different  occupations  and  economic  classes 
exhibit  different  death  rates.  We  therefore  expect  differ- 
ences in  the  items  of  qualitative  and  quantitative  series. 

However,  it  is  not  so  much  the  deviations  of  the  in- 
dividual items  from  the  average  that  interests  us  as  it  is 
their  interrelations,  the  comparison  of  their  relative  sizes. 
Thus,  the  grouping  of  the  death  rates  about  the  average, 
if  deaths  be  classified  by  occupation  or  economic  position, 
is  not  very  significant,  but  it  is  important  to  ascertain 
the  relative  mortality  of  different  occupations  and  economic 
classes,  and  to  determine  the  amount  of  difference  between 
the  individual  items.  Only  in  this  way  can  we  find  out, 
for  instance,  if  mortality  according  to  well-being  exhibits 


RELATIVE  NUMBERS  AND  AVERAGES  315 

a  definite  characteristic  conformation,  other  than  a  special 
grouping  of  the  items  about  the  average,  the  mortality 
perhaps  varying  inversely  with  economic  well-being.  In 
addition  to  these  more  theoretical  objections,  the  measure- 
ment of  the  dispersion  of  qualitative  and  quantitative  series 
is  of  little  use,  because  such  series  usually  consist  of  a 
small  number  of  items  whose  relationships  can  be  com- 
prehended without  special  computations. 

The  reason  that  qualitative  and  quantitative  series  usually 
exhibit  greater  fluctuations  than  geographical  and  time 
series  is,  therefore,  that  the  former,  as  a  rule,  originate 
from  the  use  of  a  criterion  of  a  causal  character.  If, 
for  example,  economic  position  influences  mortality,  then 
it  is  obvious  that  greater  differences  of  mortality  must 
exist  between  the  poor  and  the  rich  of  a  country  than 
between  the  total  death  rates  of  consecutive  years  during 
which  the  economic  organization  changes  but  little,  or  be- 
tween the  total  death  rates  of  different  countries  whose 
average  economic  positions  are  not  usually  as  extreme  as 
those  of  the  lowest  and  highest  classes  of  the  same  country. 
It  is  self-evident  that  as  natural  causes  operate  to  bring 
about  time  stability  and  geographical  coincidence,  so  also 
to  that  extent  they  tend  to  produce  greater  coincidence  of 
qualitative  and  quantitative  constituent  masses.  To  illus- 
trate, as  the  normal  stature  of  a  population  appears,  so 
to  say,  to  be  predetermined  by  nature,  the  various  social 
classes,  of  course,  exhibit  but  slight  differences  of  stature. 
The  influence  of  social  factors  is  greater  upon  the  normal 
length  of  life,  which  is  largely,  nevertheless,  dependent  upon 
natural  causes.  Likewise,  sex-ratios  of  children  born,  when 
found  for  qualitative  and  quantitative  constituent  masses, 
in  spite  of  the  natural  causes  at  the  basis,  exhibit  not 
unimportant  variations  in  contrast  to  the  great  stability  in 
time  and  in  geographical  distribution.  The  percentage  of 
boys  is  known  to  be  much  greater  among  stillborn  than 
among  living  births;   on  the   contrary,   it  is  less  among 


316     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

illegitimate   births   than    among   legitimate   ones    and,   at 
least  in  Germany,  less  in  large  cities  than  in  the  country," 

B.  ELEMENTARY  MATHEMATICAL  METHODS  FOR 
MEASURING  AND  REPRESENTING  THE  DISPERSION 
OF  SERIES  OF  RELATIVE  NUMBERS  AND  AVERAGES 
WHICH  CHARACTERIZE  MASSES  LIMITED  IN  A 
DEFINITE  WAY  (CONSTITUENTS  OF  A  GREATER  TO- 
TALITY) IN  SOME  OTHER  RESPECT  THAN  ACCORD- 
ING TO  MAGNITUDE 

The  elementary  mathematical  methods  by  means  of  which 
the  dispersion  of  series  belonging  to  the  third  group 
(whether  they  be  time,  space,  qualitative,  or  quantitative 
series)  can  be  measured  and  represented  are,  for  the  greater 
part,  the  same  as  those  used  for  series  consisting  of  in- 
dividual observations,  or  of  items  which  give  the  magni- 
tudes of  definite  masses  (first  and  second  groups).  The 
simplest  method  consists  in  presenting  the  extreme  mem- 
bers, maximum  and  minimum,  of  the  series  together  with 
the  average.  These  values  give  the  range  within  which 
lie  all  the  items  of  the  series.  In  addition,  the  location  of 
the  extreme  values  may  be  given.  Thus,  the  dispersion 
of  the  death  rates  for  a  series  of  years,  for  the  provinces 
of  a  country,  or  for  different  occupations,  may  be  character- 

"  Lexis  has  explained  the  slighter  excess  of  boys  among  illegiti- 
mate births  and  in  large  cities  by  suggesting  that  such  classes  may 
be  distinguished  by  different  percentages  of  premature  births  and 
has  thus  attempted  to  harmonize  these  phenomena  with  the  regula- 
tion of  the  sex  rates  by  natural  law.  (Cf.  Abhandlungen  zur 
Theorie  der  Bevolkerungs-  und  Moralstatistik,  VII,  "  The  Sex-ratio 
of  Births  and  the  Calculus  of  Probability,"  pp.  166-168.)  G.  v, 
Mayr  explains  the  greater  excess  of  boys  in  the  country  (as  with 
Jews)  by  the  fact  that  crossing  is  more  unusual  and  inbreeding  more 
common. 

The  question  whether  the  age  relation  of  the  parents,  in  the  sense 
of  the  Hofacker-Sadler  hypothesis,  has  an  influence  upon  the  sex- 
ratio  of  offspring  is,  as  yet,  undecided;  so  are  the  questions  of  the 
influence  of  fecundity  and  nourishment. 


I 


RELATIVE  NUMBERS  AND  AVERAGES  317 

ized  by  the  average  death  rate  and  the  extreme  values,  the 
year,  province,  or  occupation  to  which  the  latter  refer 
being  denoted  as  well  as  the  numerical  size  of  these  values. 

The  presentation  of  more  than  one  average,  by  which  we 
can  suitably  characterize  the  dispersion  of  series  of  the 
first  group  consisting  of  individual  observations,  is  im- 
possible for  series  of  the  third  group  and  unusual  for  those 
of  the  second  group.  If  the  presentation  of  the  extreme 
values  does  not  appear  to  be  sufficient  the  average  may 
be  supplemented  further — as  in  the  case  of  series  of  the 
first  and  second  groups — by  the  presentation  of  certain 
classes  which  possess  especial  significance. 

Under  certain  circumstances  we  may  also  consider  the 
computation  of  the  average  deviation,  which  may  be  given 
by  its  absolute  size  or  as  a  percentage  of  the  average  of 
the  series.  Such  a  computation,  however,  is  especially  diffi- 
cult for  series  of  relative  numbers  or  averages.  The  items 
of  such  series  possess  different  weights,  and  the  weights 
are  not  indicated  by  the  series.  If  we  take  the  deviation 
of  each  member  of  a  series  of  relative  numbers  or  averages 
from  the  average  of  the  series  and  then  compute  the  simple 
arithmetic  mean  of  these  deviations,  we  assume  that  the 
various  members  have  equal  weights,  which  is  usually 
contrary  to  fact.  The  assumption  of  the  equality  of  weight 
of  the  members  of  time  series  usually  introduces  but  little 
error.  For  example,  if  the  population  of  a  country  has 
changed  but  little  during  the  years  considered,  which  is 
often  the  case  for  a  short  period,  then  the  yearly  death 
or  birth  rates  may  be  considered  of  equal  weights  and  the 
average  deviation  of  death  or  birth  rates  for  the  whole 
period  may  be  computed.'^* 

'*  Thus  Adolf  Wagner  in  his  Gesetzmassigkeit  in  den  scheinbar 
willkiirlichen  Handlungen  (p.  88)  has  computed  the  mean  numerical 
deviation  of  marriage  and  birth  rates  in  order  to  determine  the 
time-movement  of  these  items  for  various  countries,  and  Mayo-Smith 
in  his  Statistics  and  Sociology   (p.  91)   has  done  the  same  for  Ger- 


318     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

In  geographical  and  qualitative  and  quantitative  series, 
however,  the  items  are,  as  a  rule,  of  such  decidedly  different 
weights  that  the  computation  of  the  simple  arithmetic 
average  of  deviations  is  not  allowable.  Imagine  a  series 
of  death  rates  for  various  provinces  or  for  various  occu- 
pations. The  individual  death  rates  are,  in  all  probability, 
of  different  weights,  as  the  provinces  are  not  equal  in  size, 
nor  do  the  occupations  contain  equal  numbers.  The  com- 
putation of  the  simple  arithmetic  average  of  the  deviations, 
by  which  it  is  assumed  that  the  death  rates  have  the  same 
weights,  would  give  an  incorrect  result.  For  example,  if 
the  larger  provinces  or  occupations  vary  but  little,  while 
the  smaller  provinces  or  occupations  vary  widely  from  the 
average,  the  average  deviation  would  be  greater  than  the 
facts  justify.  In  order  to  correctly  compute  the  mean 
deviation  one  would  have  to  take  into  consideration  the 
various  weights  of  the  deviations  of  the  series,  that  is,  pro- 
portional weights  would  have  to  be  assigned  to  the  members 
and  a  weighted  average  computed.  However,  such  a 
computation  would,  as  a  rule,  require  work  out  of  all 
proportion  to  the  value  of  the  measure  of  fluctuation 
desired. 

man  birth  rates.  Both  authors  give  the  mean  deviation  as  a  per- 
centage of  the  average  of  the  series.  The  figures  for  the  different 
months  of  a  year  may  be  considered  of  equal  weight,  if  we  are  not 
striving  for  great  accuracy,  and  their  dispersion  characterized  by 
an  index  of  fluctuation.  Thus,  Kollmann  in  his  Die  Bewegimg  der 
Bevolkerung  in  den  Jahren  1871  bis  1887  mit  Riickblicken  auf  die 
altere  Zeit  has  used  indices  of  dispersion,  computed  in  the  following 
way,  to  sum  up  the  various  degrees  of  the  monthly  variation  of  mar- 
riages, births,  and  deaths  in  the  Grand  Duchy  Oldenburg:  He 
computed  the  daily  average  for  each  month  and  for  the  whole  year; 
assuming  the  latter  to  be  1,000  he  expressed  each  average  for  a 
month  proportionately;  finally,  he  found  the  deviations  of  the  rela- 
tive numbers  for  each  month  from  1,000  and  averaged  them.  (Cf. 
Statistische  Naehrichten  iiber  das  Grossherzogtum  Oldenburg,  No.  22, 
1890,  pp.  25,  83,  114.) 


RELATIVE  NUMBERS  AND  AVERAGES  319 

C.  EXAMINATION  OF  THE  DISPERSION  OF  SERIES 
OF  NUMERICAL  PROBABILITIES  BY  THE  METHODS 
OF  THE  THEORY  OF  PROBABILITY 

The  theory  of  probability  offers  a  special  method  for 
examination  of  the  dispersion  of  series  whose  items  are  in 
the  form  of  numerical  probabilities  or  their  functions.''^'^^ 
That  is  to  say,  such  series  may,  with  reference  to  their 
dispersion,  be  compared  with  series  which  originate  from 
some  game  of  chance,  for  example,  the  drawings  of  balls 
from  an  urn  in  which  there  are  a  definite  number  of  red 
and  white  ones.  The  analogy  between  statistical  proba- 
bilities and  the  results  of  games  of  chance  already  set  forth 
(p.  171  f.),  which  is  assumed  when  the  theory  of  proba- 
bility is  applied  to  statistical  material,  may  be  summarized 

^°  Cf.  the  definition  of  numerical  probabilities,  p.  19  f. 

""  Relative  numbers  which  do  not  possess  the  character  of  numerical 
probabilities  (or  their  functions)  may  sometimes  be  treated  accord- 
ing to  the  theory  of  errors  of  observation  like  the  absolute  numbers 
originating  from  repeated  observations  of  the  same  object.  It  can 
accordingly  be  ascertained  whether  the  items  exhibit  a  grouping  cor- 
responding to  the  law  of  error;  if  this  be  the  case,  then  the  items 
may  be  considered  to  be  accidental  modifications  of  a  fixed  "  typical  " 
or  normal  value  and  the  dispersion  may  be  expressed  in  conformity 
with  the  theory  of  error  (the  "physical"  method)  by  means  of 
the  mean  or  probable  error  or  some  other  proper  measure.  Cephalic 
indices  (which  are  relative  numbers  obtained  by  taking  the  ratio 
of  the  breadth  of  the  cranium  to  its  length),  especially,  have  been 
treated  in  this  way  and  the  dispersion  has  been  proven  to  correspond 
to  the  normal  law  of  error.  We  can  thus,  perhaps,  assume  that 
the  average  cephalic  index  of  a  definite  people  possesses  a  typical 
value.  Likewise  a  generalization  of  the  law  of  error  may  be  applied 
to  relative  numbers.  Thus,  Pearson  has  applied  his  method  of  the 
"  generalized  probability  curve "  to  a  series  of  Bavarian  cephalic 
indices  and  established  an  unsymmetrical  grouping  which,  however, 
corresponds  to  an  extended  law  of  error.  Pearson  has  also  treated 
English  statistics  of  the  percent  of  paupers  by  means  of  the  "  gen- 
eralized probability  curve."  ( Cf .  "  Contributions  to  the  Mathemat- 
ical Theory  of  Evolution,"  II,  "  Skew  Variation  in  Homogeneous 
Material,"  pp.  388,  404.) 


320     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

with  especial  reference  to  the  problem  of  the  measurement 
of  dispersion  in  the  following  way: 

If  one  ball  is  drawn  from  an  urn  containing  red  and 
white  balls  in  the  ratio  6 : 4  the  objective  (theoretical) 
probability  that  a  red  one  will  appear  is  0.6,  and  that  a 
white  one  will  appear  is  0.4.  If  the  experiment  is  made 
of  drawing  a  ball  1,000  times,  replacing  it  after  each  draw- 
ing, the  percentage  of  red  or  white  balls  drawn  may  not 
exactly  coincide  with  the  objective  probability  (which 
would  be  the  case  if  600  red  and  400  white  balls  be  drawn) 
but  will  rather  closely  approximate  it.  Repeating  the  ex- 
periment and  making  note  of  the  percentage  of  red  and 
of  white  balls  in  each  drawing  of  1,000,  a  series  of  em- 
pirical probability  values  (that  is,  empirical  expressions 
for  the  objective  probability  of  the  appearance  of  a  red 
or  white  ball,  merely  affected  by  accidental  errors)  will 
be  obtained  which  group  themselves  in  a  characteristic 
manner  about  the  known  objective  (theoretical)  proba- 
bility. The  peculiarity  of  the  grouping  consists  in  the 
concentration  of  the  empirical  values  about  the  middle  of 
the  series  and  a  diminution  in  the  number  of  values  as 
their  deviation  from  the  objective  probability  increases. 
The  limits  between  which  the  empirical  values  fluctuate 
depends  essentially,  in  conformity  to  the  "  law  of  great 
numbers,"  upon  the  number  of  underlying  observations. 
The  greater  the  number  of  observations,  upon  which  the 
empirical  values  are  based  (called  the  basic  number),  the 
greater  will  be  the  precision  of  the  empirical  values,  that  is, 
so  much  closer  will  each  empirical  value  (with  a  given 
probability)  approximate  the  objective  probability;  the 
values  will  be  more  closely  grouped,  and  the  dispersion 
will  be  less.  The  smaller  the  basic  number  of  observations 
the  more  will  the  empirical  values  deviate  from  the  objec- 
tive probability  and  from  each  other.  The  amount  of  dis- 
persion of  each  of  several  series  of  empirical  values  for 
the  same  objective  probability  depends,  therefore,  upon  the 


RELATIVE  NUMBERS  AND  AVERAGES  321 

basic  number  of  observations  underlying  the  indi^ddual 
items.  From  the  objective  probability  and  the  basic  num- 
ber, the  dispersion  to  be  expected  on  the  basis  of  the  law 
of  chance  may  be  directly  computed  for  any  given  case. 
In  particular,  the  standard,  the  probable,  and  the  arithmetic 
average  deviations  of  the  empirical  values  from  the  objec- 
tive probability  (or,  as  the  case  may  be,  from  the  arithmetic 
mean  of  the  empirical  values  which  is  taken  as  the  most 
probable  value  of  the  objective  probability),  as  well  as  the 
modulus  and  the  precision  to  be  expected,  can  be  numer- 
ically determined.  The  dispersion  actually  originating  from 
a  correctly  conducted  game  of  chance  will  approximately 
coincide  with  that  theoretically  determined  by  computation. 
In  statistics  there  is  never  a  known  objective  probability 
for  the  empirical  values.  The  problem  which  statistics 
has  primarily  to  solve  is  whether  definite  values  obtained 
by  observation  (relative  numbers  in  the  form  of  probabili- 
ties or  their  functions)  may  or  may  not  be  considered,  with 
reference  to  the  dispersion  about  their  arithmetic  mean, 
to  be  empirical  values  of  some  objective  probability.  The 
objective  probability  must  be  ascertained  from  the  observa- 
tions. If  the  observed  values  arrange  themselves  about 
their  average  in  the  same  manner  and  between  the  same 
limits  as  would  empirical  probabilities  based  on  the  same 
number  of  observations  originating  from  a  correctly 
arranged  game  of  chance,  then  one  may  conclude  that  the 
statistical  observations  are  the  consequence  of  some  proba- 
bility common  to  them,  merely  affected  by  accidental  errors. 
This  conclusion  is  naturally  of  very  great  significance. 
Assuming  the  conclusion,  the  observations  lose  any  scien- 
tific significance ;  the  important  thing  being  the  theoretical 
probability  to  which  the  observations  point.  The  arith- 
metic average  of  the  observed  values  can  be  regarded  as 
the  most  probable  value  of  the  theoretical  probability.  It 
may  be  designated  as  a  **  typical  "  mean  with  reference 
to  the  grouping  of  the  items  about  it,  and  it  possesses  greater 


322     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

significance  than  any  one  of  the  empirical  items.  The  exist- 
ence of  a  theoretical  probability  signifies  that  the  phenome- 
non in  question  arises  from  natural  law  or  in  case  it  concerns 
a  social  phenomenon — for  instance,  in  its  time  fluctuations — 
that  the  general  conditions,  upon  which  the  phenomenon  de- 
pends, remain  constant,  even  if  disturbing  accidental  causes 
bring  about  small  fluctuations  so  that  the  stability  of  the 
series  is  not  complete.  The  accidental  causes  are  the  equiv- 
alent of  the  contingencies  which  arise  in  drawing  balls  from 
an  urn  (among  these  are  the  manner  in  which  the  urn 
is  shaken,  the  positions  consequently  occupied  by  the  balls 
of  different  color,  the  blind  selection  of  the  ball,  etc.)  ; 
the  constancy  of  the  general  conditions,  which  is  indicated 
by  the  stability  of  the  statistical  series,  corresponds  to  the 
constancy  of  the  ratio  between  the  balls  of  the  two  colors. 
If  the  distribution  of  a  series  of  relative  numbers,  which 
correspond  to  the  formal  conditions,  coincides  with  the 
grouping  defined  by  the  theory  of  probability,  then,  with 
Lexis,  we  may  designate  the  dispersion,  or  the  stability,  of 
this  series  as  "  normal."  If  the  relative  numbers  deviate 
from  each  other  more  than  would  be  the  case  with  a  cor- 
responding game  of  chance,  but  nevertheless  arrange  them- 
selves symmetrically  about  their  average  like  accidental 
fluctuations,  then  the  dispersion  of  the  series  may  be 
called  "  supra-normal,"  and  the  stability  "  infra-normal  "; 
if  the  relative  numbers  deviate  less  from  each  other  than 
do  the  items  of  the  corresponding  accidental  series  then 
the  dispersion  of  the  series  is  "  infra-normal,"  its  stability 
is  *' supra-normal." '^ 

"  Cf.  Lexis,  Theorie  der  Massenerscheinungen,  p.  34,  and  article 
"  Gesetz "  in  the  Handw.  d.  Staatsw.  The  expressions  "  normal  " 
dispersion  and  "  normal "  stability  must,  naturally,  not  be  under- 
stood to  mean  that  such  dispersion  and  stability  are  the  rule  in 
statistics.  Such  distribution  is  to  be  normally  expected  in  the  domain 
of  games  of  chance,  the  peculiar  domain  of  the  theory  of  probability, 
but  it  is  a  rare  exception  in  the  field  of  statistics. 


RELATIVE  NUMBERS  AND  AVERAGES  323 

The  methods  by  means  of  which  one  may  determine 
whether  the  dispersion  of  a  statistical  series  of  relative 
numbers  of  the  correct  form  coincides  with  the  dispersion 
which  would  result  from  a  game  of  chance  conforming  to 
the  theory  of  probability  were  developed  mainly  by  Lexis.''® 
Westergaard,  von  Bortkiewicz,  Czuber,  and  Blaschke  have 
also  worked  in  this  field.  The  object,  particularly,  is  to 
determine  the  values  which  would  be  expected  according 
to  the  theory  of  probability  and  to  compare  with  them  the 
grouping  of  the  values  obtained  by  observation.  Instead 
of  comparing  the  entire  distribution  of  the  values  obtained 
by  observation  and  by  theory,  a  simpler  method  originated 
by  Lexis  may  be  applied.  This  method  consists,  first,  in 
computing  the  probable  error, which  gives  theoretically  (that 
is,  in  conformity  to  the  theory  of  probability)  a  constant 
probability  for  the  empirical  values  resting  upon  corre- 
sponding numbers  of  observations  (computation  of  the 
probable  error  according  to  the  ''  combinational,"  "  statis- 
tical," or  "  indirect  "  method),  and  second,  in  determining 
the  probable  error  in  the  sense  of  the  theory  of  errors 
of  observation  directly  from  the  observed  values,  whereby 
the  items  are  considered  simply  to  be  consequences  of  some 
measurement  subject  to  accidental  errors,  without  reference 
to  the  question  whether  these  values  may  or  may  not  be 
considered  as  numerical  probabilities  (determination  of 
the  probable  error  according  to  the  "  physical  "  or 
**  direct  "  method).  If  the  probable  errors  (or  the  medial 
or  average  errors,  moduli,  or  precisions,  as  one  may  select) 
computed  by  the  two  methods  coincide,  that  is,  if  their 

"  Cf.  especially  "  Das  Geschlechtsverbiiltnis  der  Geborenen  imd  die 
Wahrscheinlichkeitsrechnung  "  and  "  Uber  die  Theorie  der  Stabilitat 
statistischer  Reihen  "  (which  first  appeared  in  the  Hildebrand-Conrad 
Jahrbucher,  1876  and  1879,  but  are  now  contained  in  Abhand- 
lungen  zur  Theorie  der  Bevulkerungs-  und  Moralstatistik)  as  well 
as  Zur  Theorie  der  Massenerscheinungen  in  der  menschlichen  Ge- 
sellschaft  (1877)  and  the  article  "  Geschlechtsverhiiltnis  der  Ge- 
borenen und  Gestorbenen  "  in  the  Handw.  d.  Staatsw, 


324     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

quotient  "*  approximates  1,  this  shows  that  the  deviations 
of  the  items  from  the  mean  are  of  the  same  kind  as  those 
resulting  from  a  game  of  chance  constructed  to  correspond 
and  that  these  items  may  actually  be  considered  empirical 
values  of  some  constant  probability.  If  the  probable  error 
computed  according  to  the  physical  method  is  greater  than 
that  given  by  the  combinational  method  then  the  dispersion 
of  the  series  is  supra-normal,  if  less,  it  is  infra-normal.^^'^** 
As  a  matter  of  fact,  series  of  statistical  relative  numbers 
which  may  be  considered  numerical  probabilities  or  their 
functions  and  which  exhibit  normal  dispersion  correspond- 
ing to  the  theory  of  probability  occur  but  rarely.  The  best 
known  illustration  is  the  sex-ratio  of  children  born,  which 
Lexis  has  demonstrated,  upon  the  basis  of  Prussian,  Eng- 
lish, and  French  statistics,  to  be  subject  to  no  greater 
variation  both  in  its  space  and  time  distribution — at  least 
for  the  same  country  and  within  a  moderate  space  of  time — 
than  is  to  be  expected  from  the  assumption  of  a  constant 

^'a  Designated  "  divergency-coefficient "  by  Dormoy  and  "  error-rela- 
tion "  by  V.  Bortkiewicz. 

'°  For  a  more  inclusive  treatment  see  Lexis,  Abhandlungen  zur 
Theorie  der  Bevolkerungs-  und  Moralstatistik,  VII,  "  The  Sex-ratio 
of  Children  Born  and  the  Theory  of  Probability,"  p.  153  S. 

'"  Series  of  averages  may  be  treated  according  to  the  methods  of 
the  theory  of  probability  in  the  same  way  as  series  of  relative  num- 
bers. The  essential  question  is,  Can  we  consider  these  averages, 
with  reference  to  their  dispersion,  as  accidental  modifications  of  some 
base  value?  If  a  comparison  of  the  actual  with  the  expected  pre- 
cision shows  the  existence  of  a  supra-normal  dispersion,  then  it  fol- 
lows that  the  values  under  investigation  are  subject  to  a  time  or 
place  fluctuation.  Anthropometric  and  meteorological  averages,  espe- 
cially, are  investigated  in  such  a  manner.  (Cf.  L.  v.  Bortkiewicz, 
"  Kritische  Betrachtungen  zur  theoretischen  Statistik,"  Jahrb.  f.  Nat. 
u.  Stat.,  3rd  series.  Vol.  X  (1895),  p.  342  f.,  and  "  Anwendungen  der 
Wahrscheinlichkeitsrechnung  auf  Statistik  "  in  the  Enzyklopadie  der 
mathematischen  Wissenschaften,  p.  835  f.,  as  well  as  Czuber,  Die 
Wahrscheinlichkeitsrechnimg  und  ihre  Anwendung  auf  Fehleraus- 
gleichung,  Statistik  und  Lebensversicherung,  p.  341  f.) 


■ 


RELATIVE  NUMBERS  AND  AVERAGES  325 

probability  for  the  birth  of  a  boy  or  girl.^^  Differences 
in  the  degree  of  probability  exist,  however,  for  certain 
qualitative  and  quantitative  subdivisions.  Thus,  we  find 
that  in  most  countries  there  is  a  smaller  surplus  of  boys 
among  illegitimate  births  than  among  legitimate  births,  and 
a  greater  surplus  of  boys  among  the  stillborn  than  among 
the  living-born.^- 

But  the  special  sex-ratios  of  the  illegitimate  and  the 
stillborn  give  time  series  of  normal  dispersion,  just  as  the 
sex-ratio  based  upon  all  births  does  and,  therefore,  possess 
independent  typical  character.  That  there  is  a  constant 
total  probability  for  the  birth  of  a  girl  or  a  boy  (with 
no  differentiation  of  legitimate  births  from  illegitimate,  or 
still-births  from  living),  in  spite  of  the  above  considera- 
tions, is  due  to  the  fact  that  the  percentage  of  illegitimate 
births  or  stillborn  fluctuates  but  little.  Aside  from  the 
two  classes  named  there  are  probably  other  qualitative  or 

*^  Cf.  Geissler,  "  Beitrage  zur  Frage  des  Geschleehtsverhiiltnissea 
der  Geborenen"  (Zeitschr.  d.  kgl.  sachs.  stat.  Bur.,  XXXV,  1889, 
Kos.  1  and  2 ) ,  and  Lehr,  "  Zur  Frage  der  Walirscheinlichkeit  von 
weiblichen  Geburten  und  von  Totgeburten"  (Zeitschr.  f.  d.  ges, 
Staatsw.,  XLV,  1889,  pp.  172  ff.  and  524  flf.),  and  Stieda,  Das  Sexual- 
verhiiltnis  der  Geborenen  (1875).  For  Austria,  Czuber  (Wahrschein- 
lichkeitsreehnung,  pp.  325-328)  has  shown  that  a  moderate  supra- 
normal  dispersion  holds  for  the  relative  frequency  of  a  male  birth 
among  living  births  for  the  period  1866-1897.  Czuber  does  not 
ascribe  this  fact  to  a  time  change  of  fundamental  conditions  but 
to  the  heterogeneity  due  to  the  contribution  of  many  races  to  the 
data  used.  In  the  less  extensive  categories  of  legitimate  and  illegiti- 
mate still-births  the  dispersion  is  approximately  normal.  In  Eng- 
land and  Wales  the  fluctuation  of  the  sex-ratio  appears  to  be  con- 
tinually affected  by  a  definite  evolutionary  tendency;  according  to 
the  Report  of  the  Registrar  General  for  the  year  1893  (p.  xxviii) 
the  excess  of  boys  has  gradually  decreased  from  105.4  and  105.0  in 
the  years  1844  and  1845  respectively,  to  below  104  in  the  year  1893, 
and  it  has  not  been  as  high  as  104.0  since  1885.  Upon  the  influence 
of  race  upon  the  sex-ratio  see  the  note  on  p.  312,  above. 

'^  Cf.  the  dissertation  of  Stark  on  the  Geschlechtaverhiiltnis  bei 
xinehelichen  Geburteu  und  bei  Totgeburten  (Freiburg,  1877). 


326     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

quantitative  special  classes  of  births  that  possess  inde- 
pendent probabilities  for  the  birth  of  a  boy  or  a  girl. 
Thus,  according  to  Geissler's  researches  based  on  statistics 
of  Saxony,  more  or  less  fruitful  marriages  are  such  classes ; 
likewise,  the  influence  of  the  residence  (whether  in  city 
or  country),  the  ages  of  the  parents,  the  nourishment, 
etc.,  are  perhaps  not  without  influence.  But  all  of  these  dif- 
ferences may  persist  in  like  proportions,  so  that  a  constant 
total  probability  results  for  the  whole  population.^^ 

Aside  from  the  sex-ratios  of  births,  Lexis  has  shown 
that  there  is  normal  dispersion  among  the  sex-ratios  of 
deaths  occurring  in  the  lowest  age  classes,  and  to  a  degree 
also  among  those  occurring  in  the  highest  age  classes — 
but  not  among  those  in  the  middle  classes.  Consequently, 
for  such  age  classes,  in  which  physiological  conditions 
are  of  primary  influence  on  mortality,  we  may  speak 
of  a  constant  probability  for  the  deaths  of  the  two 
sexes.  This  evidently  indicates  a  constant  cause-com- 
plex, especially  relating  to  the  mortality  of  childhood, 
in  consequence  of  which  a  definitely  greater  percent- 
age of  boys  die  than  girls.  We  may  then  agree  with 
Lexis*  thesis  "  that  the  average  resistance  of  boys 
to    death    is,    upon    organic    grounds,    less    by    a   fixed 

"*  Lexis  in  the  article  "  Geschlechtsverhaltnis  der  Geborenen  "  in 
the  Handw.  d.  Staatsw.,  2nd  ed.,  p.  180;  cf.  also  "Das  Geschlechts- 
verhaltnis der  Geborenen  und  der  Wahrscheinliehkeitsrechnung "  in 
Lexis'  Abhandlungen  and  the  bibliography  attached  to  the  article 
cited  above.  Lexis  has  investigated  (Theorie  der  Massener- 
scheinungen,  pp.  74-78)  the  special  relation  existing  among  twin 
births  and  has  found  that  the  ratio  of  the  number  of  twins  in 
which  both  were  boys  to  the  number  of  twin  (both)  girls  is  the 
same  as  the  ratio  between  single  male  and  female  births.  The 
dispersion  of  these  ratios  for  the  older  divisions  of  Prussia  during 
the  period  1862-1873  was  very  near  normal,  as  was  also  the  case 
for  the  percentage  of  mixed  twin  births.  Aside  from  Lexis  the  sex- 
ratios  among  multiple  births  have  been  investigated  by  Westergaard, 
Geissler,  Neefe,  and  Herri, 


1 


RELATIVE  NUMBERS  AND  AVERAGES  327 

amount  than  that  of  girls."**  Further,  the  sex-ratios  of 
the  survivors  of  one  and  the  same  generation  at  the  end 
of  the  first  years  of  life  can  be  cited  *^  as  likewise  possessing 
normal  dispersion  and  a  typical  value.  It  is  to  be  noted 
that  the  relative  numbers  cited  are  all  secondary  (analyt- 
ical, relative)  numerical  probabilities.  Also  the  other  cases 
of  normal  dispersion  which  can  be  established — even  though 
merely  for  single  countries  and  for  definite  periods  of 
time — mostly  relate  to  secondary  probabilities.  Thus,  the 
percentage  of  females  condemned  for  larceny  (in  relation 
to  the  total  number  condemned  for  that  reason),  which 
according  to  Westergaard  ®®  exhibited  a  normal  dispersion 
in  Denmark  during  the  period  1867-1885,  represents  a 
secondary  probability.  Other  illustrations  are  the  ratio  of 
female  suicides  to  the  total  number,  which  Westergaard^'^ 
found,  in  Denmark  for  the  period  1861-1886  and  in  Belgium 
for  the  period  1865-1883,  to  coincide  with  the  experience 
of  games  of  chance,  the  relative  frequency  of  suicide  by 
hanging,  and  the  relative  frequency  of  suicide  in  the 
months  October,  November  and  December,  for  both  of  which 
Westergaard  found  normal  stability  for  the  Danish  figures, 
1861-1886.®*     In  other  countries  the  percentage  of  female 

**  Abhandlungen,  p.  204.  Cf.  Geigel,  Die  Stabilitat  des  Geschlechts- 
verhaltnisses  der  Gestorbenen,  1880. 

*'  W.  Kammann,  Das  Geschlechtsverhaltnis  der  Uberlebenden 
in  den  Kinderjahren  als  selbstandige  massenphysiologiscbe  Kon- 
stante,  Gottingen,  1900  (abstract  in  the  Jabrb.  f.  Nat.  und  Stat.,  3rd 
series,  Vol.  XIX  (1900),  p.  382  f.). 

"  Grundzuge  der  Theorie  der  Statistik,  p.  50. 

*'  See  loc.  cit.,  p.  44  f. 

*' Grundzuge,  pp.  45  f.,  47.  v.  Bortkiewicz  finds  ("  Kritische 
Betrachtungen  zur  theoretischen  Statistik,"  Conrad's  Jahrb.,  3rd 
series,  Vol.  VIII  (1894),  p.  672),  in  opposition  to  Westergaard,  that 
the  dispersion  of  the  frequency  of  suicides  by  hanging  (Wester- 
gaard's  illustration)  departs  from  theory  by  a  not  inconsiderable 
amount.  Such  a  difference  of  opinion  is  possible  because  in  judging 
the  dispersion  of  a  series  from  the  standpoint  of  the  theory  of 
probability    a    certain    subjective    element   always    enters,    v.    Bort- 


328        DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

suicides  and  the  classes  of  suicides  according  to  the  manner 
of  death  possess  a  smaller  degree  of  stability.  Lexis  has 
shown,  for  instance,  that  in  France  the  percentage  of 
suicides  by  drowning  exhibited  a  tendency  to  decrease  for 
both  sexes  during  the  period  1835-1868,  and  that  there  is 
normal  stability  only  for  the  female  sex  and  that  only  for 
a  part  of  this  period  selected  for  its  small  fluctuations. 
Lexis  has  further  shown  that  the  relative  number  of  female 
suicides  exhibited  a  tendency  to  decrease  during  that 
period.^^ 

The  most  thoroughly  investigated  "  primary  "  numerical 
probability  in  the  demographic  field  is  the  probability  of 
death.  This  probability  for  the  years  of  childhood  doubt- 
less presents  a  decided  supra-normal  dispersion  in  its  time 
fluctuations,®"  while  from  the  age  of  ten  years  and  up  it 
presents  a  greater  stability.®^  J.  H.  Peek  ^^  found  dis- 
persion differing  but  little  from  the  normal  (slightly  supra- 
normal)  for  the  male  population  of  the  Netherlands.  A 
very  good  approximation  to  normal  dispersion  was  found 
by  Peek  in  the  mortality  ratios  which  have  been  deduced 
from  the  Dutch  civil  service  statistics  (1878-1894)  for  the 
construction  of  the  first  civil  service  mortality  table  for 

kiewicz  has,  however,  classified  Westergaard's  data  of  the  frequency 
of  suicide  by  hanging  according  to  the  sex  of  the  suicide  and  has 
found  that  the  dispersion  for  each  sex  is  apparently  normal,  the 
dispersion  for  the  female  sex  being  unquestionably  so. 

"  Cf.  Theorie  der  Massenerscheinungen,  pp.  83-87,  and  article 
"  Gesetz  "  in  Handw.  d.  Staatsw.,  2nd  ed.,  p.  239. 

»"  Cf.  Lewis,  ibid.  pp.  78-82. 

'•  Tschuprow  ("Die  Aufgaben  der  Theorie  der  Statistik," 
Schmoller's  Jahrbuch,  1905,  p.  54  f. )  ascribes  the  smaller  degree  of 
stability  of  the  mortality  of  childhood  as  compared  with  that  of 
later  years  of  age  to  the  greater  inlluence  that  climatic  conditions 
have  upon  the  health  of  delicate  children  as  compared  -with  their 
influence  upon  adults  and  to  the  greater  susceptibility  of  children  to 
epidemics. 

^^  "  Das  Problem  des  Risiko  in  der  Lebensversicherung,"  Zeitschrift 
fiir  Versicherungsrecht  und  -Wissenschaft,  V   (1899),  pp.  169-197. 


RELATIVE  NUMBERS  AND  AVERAGES  329 

the  Netherlands.^^  Normal  dispersion  and,  therefore,  the 
maximum  stability  of  mortality  ratios  have  been  sho\vn 
to  exist  among  insured  persons,  by  G.  Bohlmann  °*  upon 
the  basis  of  the  observations  of  the  insurance  societies  of 
Gotha  and  Leipsic,  and  by  E,  Blaschke  ^^  upon  the  basis  of 
the  observations  underlying  the  mortality  table  of  the 
twenty  British  Societies  from  the  year  1869.  E.  Blaschke 
has  demonstrated  the  normal  dispersion  of  the  frequency 
of  invalidity  between  the  ages  of  20  and  55  (upon  the 
basis  of  Zimmerman's  statistics  of  the  invalidity  of  em- 
ployees of  German  railways  and  Kaans's  statistics  of  Aus- 
trian miners'  friendly  societies ).^° 

Experience  shows  tliat  other  series  of  relative  num- 
bers based  upon  demographic  or  moral-statistical  data  do 
not,  as  a  rule,  possess  normal  dispersion,  whether  the  items 
are  for  different  years  or  for  different  geographical  areas. 
Many  such  series  have  a  definite  evolutionary  tendency,  or 
exhibit  either  periodic  fluctuations,  or  non-periodic  varia- 
tions caused  from  time  to  time  by  special  economic  or 
political  events,  all  of  these  fluctuations  being  in  evident 
contradiction  to  the  theory  of  probability.^^     Even  where 

•'  Cf.  Czuber,  Wahrscheinliclikeitsrechnung,  p.  333. 

"*  Uber  angewandte  Mathematik,  Leipzig,  1900,  p.  142. 

*°  Cf.  Die  Anwendbarkeit  der  Wahrscheinlichkeitslehre  im  Ver- 
sicherungswesen,  Vienna,  1901,  and  Vorlesungen  iiber  mathematische 
Statistik,  1906,  p.  144. 

°°  Vorlesungen  iiber  mathematische   Statistik,  p.   144   f. 

'^  For  series  or  parts  of  series  \Yhich  exhibit  a  very  slight  tend- 
ency to  increase  or  decrease  we  may  overlook  this  tendency  and  pro- 
ceed from  the  assumption  of  a  constant  probability.  We  may  also 
attempt  to  explain  an  evolutionary  series  by  assuming  a  probability 
which  varies  in  a  definite  manner  (increasing  or  decreasing),  and 
we  may  measure  the  deviation  for  a  given  year  from  the  hypothetical 
probability  assumed  for  that  year.  'Lexis,  who  has  discussed  this 
point  (Theorie  der  Massenerscheinungen,  p.  32;  Abhandlungen,  p. 
192  f.),  holds  that  the  utility  of  such  a  proceeding  is  very  doubtful. 
The  accepted  norm  for  the  variation  of  the  objective  probability 
must  always  be  arbitrary.     If  we  represent  this  variation  by  means 


330     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

there  is  no  definite  evolution  and  no  periodic  or  other  wave 
movement,  the  condition  of  the  symmetric  distribution  of 
the  items  about  the  average  is  frequently  not  satisfied. 
Finally,  in  those  cases  where  this  condition  is  satisfied  and 
the  items  are  distributed  about  their  average  like  accidental 
fluctuations  the  dispersion  of  the  series  is  usually  "  supra- 
normal  "  and  the  stability  *'  infra-normal,"  that  is,  the 
deviations  from  the  average  are  greater  than  those  called 
for  by  the  theory  of  probability,  and  the  probable  or  the 
mean  error  computed  according  to  the  "  physical  "  method 
is  often  several  times  greater  than  the  corresponding  error 
computed  by  the  "  combinational  "  method,  the  precision 
computed  by  the  former  method  is  considerably  less  than 
it  should  be,  theoretically.  Series  of  infra-normal  dis- 
persion, that  is,  series  in  which  the  actual  probable  or  mean 
deviation  from  the  average  is  less  and  the  actual  precision 
is  greater  than  that  to  be  expected  by  the  theory  of  proba- 
bility, have  up  to  this  time  not  been  discovered.  We  may 
assume  that  such  an  infra-normal  dispersion  or  supra- 
normal  stability  could  be  exhibited  only  by  mass-phenomena 
governed  by  some  exterior  restricting  force.^^  However, 
it  is  to  be  remembered  in  this  connection  that  the  theory 
of  probability  sets  a  very  strict  standard  where  great  num- 
bers of  observations  are  given.  Series  which,  according  to 
the  theory  of  probability,  possess  a  decidedly  supra-normal 
dispersion,  may  appear  to  be  extraordinarily  stable  to  the 
non-mathematical  statistician,  who  lacks  a  standard  depend- 
ing strictly  upon  the  number  of  observations.  Thus,  the 
numerous  series  which,  to  certain  writers,  seemed  to  prove 
**  regularity  "  and  "  law  "  among  statistical  phenomena, 

of  a  curve  or  a  broken  line,  then  it  is  easy  to  draw  the  line  or 
curve  so  that  the  deviations  of  the  observed  numbers  from  it  are 
a  minimum.  It  would  be  least  objectionable  to  explain  series  having 
a  uniform  development  as  time  progresses  through  assuming  a 
probability  likewise  changing  uniformly;  such  regular  evolutionary 
series  do  not,  however,  actually  occur. 

•*  Cf.  Czuber,  Wahrscheinlichkeitsrechnung,  p.  322. 


m 


RELATIVE  NUMBERS  AND  AVERAGES  331 

have  been  most  largely  those  series  which,  according  to  the 
theory  of  probability,  do  not  possess  normal  dispersion. 
On  the  other  hand,  with  a  small  number  of  observations 
the  theory  of  probability  allows  for  deviations  of  consider- 
able size  which  are  to  be  looked  at  as  purely  accidental,  yet 
which  may,  perhaps,  appear  so  considerable  to  the  non- 
mathematical  statistician  as  to  require  a  special  expla- 
nation. 

Series  of  relative  numbers  in  which  the  items  are  dis- 
tributed symmetrically,  yet  with  a  supra-normal  disper- 
sion, are  frequent. ^^  Lexis,^''"  who  is  supported  by  other 
mathematical  statisticians,^"^  explains  such  series  by  the  as- 
sumption that  the  items  are  the  result,  not  of  a  single  con- 
stant probability  as  in  the  case  of  normal  dispersion,  but  of  a 
variable  probability  which  is  itself  subject  to  accidental 
fluctuations.  Series  of  supra-normal  dispersion  may,  per- 
haps, be  compared  to  the  results  of  a  game  of  chance,  in 
which  red  and  white  balls  are  drawn  from  several  urns, 
which  do  not  contain  red  and  white  balls  in  the  same,  but 
in  varying  ratios,  those  ratios  being  themselves  affected  by 
accidental  errors.  There  is  a  theoretical  probability  recog- 
nizable even  in  such  series  which,  however,  varies  acciden- 
tally about  a  mean  from  one  series  of  observations  to  an- 
other (which  may  relate  to  years,  months,  provinces,  etc.). 

The  supra-normal  dispersion  of  these  series  results 
because  two  different  causes   of  errors  are  in  operation, 

"  Thus  Lehr  has  shown  that  in  Germany  during  1841-1885  the 
ratio  of  the  number  of  still-births  to  the  total  number  of  births, 
as  well  as  the  ratio  of  the  number  of  deaths  to  the  total  living 
population,  exhibits  deviations  from  the  mean  that  are  to  be  con- 
sidered accidental  disturbances,  although  the  dispersion  of  the  single 
items  is  considerably  supra-normal.  (Lexis,  article  "  Gesetz "  in 
Handw.  d.  Staatsw.,  Vol.  V,  p.  239.) 

100  (-If  Theorie  der  Massenerscheinungen,  especially  p.  26,  and 
Abhandlungen,  pp.  135  f.,  176-184. 

^"^  Cf.,  for  example,  v.  Bortkiewicz,  Das  Gesetz  der  kleinen  Zah- 
len,  §  14. 


332     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

first,  the  "  combinational  "  or  "  statistical  "  cause  of 
errors,  which  is  also  connected  with  a  constant  prob- 
ability and  causes  the  deviations  of  the  empirical  values 
from  the  theoretical  probability;  second,  the  "  physical  " 
or  "  physiological  "  cause  of  errors,  which  originates 
from  the  fluctuations  of  the  objective  probability.  These 
two  causes  of  errors  correspond  to  two  fluctuation  com- 
ponents, which  Lexis  calls  respectively  "  normal-acci- 
dental "  or  "  unessential,"  and  the  "  physical  "  com- 
ponents, while  von  Bortkiewicz  calls  them  the  '*  normal 
error  "  and  the  "  absolute  excess  of  error."  These  two 
components  cooperate  in  producing  the  total  deviations, 
which  are  found  by  direct  observation  of  the  fluctua- 
tions. 

If  the  dispersion  or  the  stability  of  several  series  are  to 
be  compared,  then,  according  to  Lexis,  only  the  physical 
fluctuation  component  is  of  essential  importance.  Series 
of  normal  dispersion  (whose  deviations  from  the  mean 
depend  exclusively  upon  the  "  normal-accidental  "  com- 
ponent) may  indeed  possess  various  degrees  of  variation, 
but  this  difference  is  the  exclusive  consequence  of  differ- 
ences in  the  probability  and  in  the  number  of  observations, 
and  the  series  compared  coincide  as  to  the  constancy  of  their 
theoretical  probability  and  the  non-existence  of  a  physical 
fluctuation  component.  Series  of  supra-normal  dispersion, 
on  the  contrary,  depend  upon  the  simultaneous  operation 
of  the  "  normal-accidental  "  and  a  ''  physical  "  compo- 
nent of  fluctuation.  In  order  to  compare  two  such  series 
we  must,  according  to  Lexis,  eliminate  the  normal-acciden- 
tal components  in  both  series  from  the  values  containing  the 
total  fluctuations  and  merely  regard  the  physical  fluctuation 
components  which  are  independent  of  the  number  of  ob- 
servations, as  it  is  only  in  these  components  that  the  degree 
of  the  variability  of  the  respective  probabilities  of  the  series 
compared  is  expressed,  that  is,  how  great  are  the  fluctua- 
tions to  which  these  probabilities  are  subjected. 


RELATIVE  NUMBERS  AND  AVERAGES  333 

It  has  been  established  by  Lexis  ^"^  that  approximately 
normal  dispersion  is  exhibited  especially  by  series  of  relative 
numbers,  which  are  based  on  only  a  moderate  basic  number 
of  observations  and  that  the  stability  of  statistical  relative 
numbers,  as  a  rule,  varies  inversely  with  this  basic  number. 
If  the  basic  number  is  reduced  by  means  of  a  time,  space, 
or  qualitative  or  quantitative  subdivision  of  the  statistical 
material  it  is  not  seldom  that  an  unquestionably  supra- 
normal  dispersion  is  changed  to  an  approximately  normal 
one.  This  phenomenon  may  have  two  causes.  It  doubtless 
rests  essentially  upon  the  method  of  measuring  dispersion, 
used  by  the  mathematical  statisticians,  but,  on  the  other 
hand,  it  may  to  a  certain  extent,  as  will  be  described  later, 
depend  upon  the  lack  of  interdependence  of  the  ele- 
ments belonging  to  a  statistical  mass,  and  the  fact  that 
these  elements  are  influenced  by  "  causes  operating  con- 
jointly." 

The  influence  of  the  method  of  measuring  dispersion 
asserts  itself  in  the  following  way:  The  measurement  is 
effected — as  explained  above — by  comparison  of  the  values 
obtained  for  the  probable  or  mean  error  (or  the  precision) 
computed,  first,  by  the  ' '  physical  ' '  method  and,  second,  by 
the  "  combinational  "  method.  If  the  value  found  by  the 
physical  method  is  essentially  greater  than  the  other,  then 
the  dispersion  is  supra-normal  in  a  degree  determined  by 
the  difference  between  the  values.  In  case  of  supra-normal 
dispersion  the  first  value  consists  of  two  components,  the 
normal-accidental  and  the  physical,  while  the  second  value 
has  but  one  fluctuation  component,  i.  e.,  that  corresponding 
to  the  normal-accidental  component.  The  difference  be- 
tween the  probable  or  mean  errors  (or  the  precision  values) 
computed  by  the  two  methods  depends,  therefore,  upon 
the  ratio  between  the  two  fluctuation  components  named. 
Now  the  normal-accidental  component,  in  conformity  to  the 

"^  Cf.  Abhandlungen,  VIII,  "  Concerning  the  Theory  of  the  Sta- 
bilitj'  of  Statistical  Series,"  p.  187  f. 


334     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

law  of  great  numbers,  varies  inversely  with  the  basic  num^ 
ber  of  observations,  while  the  physical  component  is,  in 
general,  independent  of  the  basic  number.  Therefore,  with 
relatively  great  basic  numbers  the  physical  fluctuation  com- 
ponent prevails,  while  with  relatively  small  basic  numbers 
the  normal-accidental  component  predominates.  Conse- 
quently, if  the  accidental  fluctuations,  to  which  the  objective 
probability  underlying  the  series  is  subject,  are  not  great, 
then  with  a  moderate  basic  number  the  physical  fluctuation 
component  is  almost  wholly  concealed  and  computation 
gives  approximately  normal  dispersion,  while  the  same 
statistical  phenomenon  founded  upon  greater  basic  num- 
bers gives  a  decided  supra-normal  dispersion. 

From  the  greater  constancy  of  relative  numbers  with 
moderate  sized  basic  numbers  it  does  not  follow,  however, — 
as  was  emphasized  especially  by  von  Bortkiewicz,^"^ — as  a 
postulate  of  statistical  research  that  we  should  be  content 
with  small  basic  numbers  and  base  our  conclusions  upon 
the  resulting  statistical  material.  "  It  is,  on  the  con- 
trary, of  greater  statistical  interest  to  establish  the  physical 
fluctuation-component,  which  is  obscured  by  the  use  of  small 
basic  numbers.  For  its  numerical  magnitude  is  a  measure, 
independent  of  the  operation  of  "  accidental  causes,"  of 
the  time  changes  which  are  experienced  by  the  probability 
in  question.  At  the  same  time  it  is  to  be  observed  whether 
the  latter  increases  or  decreases  with  time.  But  the  amoimt 
and  direction  of  the  variation  will  be,  in  so  far  as  it  is  a 
question  of  eliminating  the  accidental,  so  much  more  cer- 
tainly ascertainable  the  more  numerous  are  the  observa- 
tions upon  which  the  relative  numbers  rest.  Therefore,  a 
limitation  of  the  number  of  observations  is  not  advisable. 
However,  an  investigation  of  statistical  series  with  smaller 
basic  numbers  will  recommend  itself  on  account  of  those 
general  theoretical  interests  which  are  concerned  in  demon- 

108  « ipjjg  Theory  of  Population  and  Moral  Statistics  According  to 
Lexis,"  Jahrb.  f.  Nat.  u.  Stat.,  Vol.  XXVU  (1904),  p.  239. 


RELATIVE  NUMBERS  AND  AVERAGES  335 

strating  cases  of  an  approximately  normal  stability. ' '  The 
statistician  is,  therefore,  correct  in  generally  attempting  to 
utilize  great  masses  of  observations,  i7j  which  the  accidental 
errors  are  eliminated,  so  that  the  non-accidental  fluctuations 
or  changes  during  the  course  of  time  may  be  established. 
If  we  desire,  however,  to  investigate  the  relation  of  the 
law  of  chance  to  statistical  data,  that  is,  to  determine 
whether  the  propositions  and  methods  of  the  calculus  of 
probability  are  applicable  to  statistics,  then  the  conditions 
of  the  investigation  must  be  so  shaped  that  the  effects  of 
the  factor,  which  is  to  be  studied,  are  possible  of  evalua- 
tion.^«* 

From  this  point  of  view  von  Bortkiewicz  has  continued 
the  investigations  of  Lexis  concerning  the  question  of  the 
greater  constancy  of  relative  numbers  with  moderate  sized 
basic  numbers.  He  obtained  important  conclusions  and 
showed,  in  particular,  that  statistical  series  consisting  of 
very  small  absolute  numbers  often  exhibit  stability  almost 
completely  satisfying  the  requirements  of  the  theory  of 
probability,  and  named  this  phenomenon  ' '  the  law  of  small 
numbers"  ("  Gesetz  der  kleinen  Zahlen  ").  As  illus- 
trating this  law  von  Bortkiewicz  used  certain  figures  from 
suicide  and  accident  statistics,  figures  for  consecutive  years 
representing  very  small ' '  numbers  of  events, ' '  while,  at  the 
same  time,  each  item  originated  from  a  great  number  of 
observations  (that  is  to  say,  men  under  observation).  The 
first  illustration  was  based  on  the  suicides  of  Prussian 
children  under  ten  years  during  the  period  1869-1893.  The 
annual  number  of  suicides  of  boys  fluctuated  between  0  and 
6.  For  girls  there  were  some  years  with  no  suicides ;  there 
was  but  one  year  in  which  more  than  one  suicide  was  com- 
mitted, and  in  that  year  there  were  but  two  suicides.  There 
was  coincidence  between  the  actual  dispersion  found  and 
the  dispersion  theoretically  expected  for  boys  and  girls 
alone,  as  well  as  for  both  sexes  combined.     The  second  illus- 

*•*  Cf.  V.  Bortkiewicz,  Gesetz  der  kleinen  Zahlen,  preface,  p.  v. 


336     DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

tration  is  the  female  suicides  in  the  small  German  States 
of  Schaumburg-Lippe,  Waldeck,  Liibeck,  Reuss  a.  L., 
Lippe,  Schwartzburg-Rudolstadt,  Mecklenburg-Strelitz  and 
Schwartzburg-Sondershausen  during  the  period  1881-1894. 
The  annual  number  of  suicides  in  these  states  fluctuated 
between  0  and  10  and  the  dispersion  agreed  with  the  theo- 
retical distribution  computed  a  priori.  Similar  results  were 
given  by  the  statistics  of  accidental  deaths  among  the  em- 
ployees in  eleven  German  corporations  during  the  years 
1886-1894  (the  number  fluctuated  between  0  and  14,  the 
numbers  over  10  occurring  each  but  once),  and  by  the 
number  killed  in  the  German  army  by  being  kicked  by  a 
horse  during  the  period  1875-1894. 

This  remarkable  constancy  of  small  numbers  of  events, 
as  well  as  the  constancy  of  relative  numbers  based  upon 
small  numbers  of  observations,  doubtless  essentially  depends 
upon  the  method  used  to  measure  dispersion,  which  tends 
to  conceal  the  physical  fluctuation  component  of  a 
variable  probability  if  a  small  number  of  observations  or 
events  are  given.  Von  Bortkiewicz  has,  however,  an  ex- 
planation of  this  constancy  independent  of  the  method  of 
measuring  dispersion  in  his  theory  of  "  causes  operating 
conjointly. ' ' 

In  all  investigations  by  means  of  the  theory  of  probability 
we  proceed  from  the  assumption  that  the  events  are  wholly 
independent  of  each  other.  This  assumption  is,  however, 
a  circumstance  not  always  realized.  To  this  fact  is  due  the 
considerable  difference  that  is  so  often  observed  in  statis- 
tics between  the  actual  and  the  expected  (theoretical) 
groupings.  But  the  same  fact,  with  the  help  of  the  theory  of 
"  conjointly  operating  causes,"  also  explains  the  greater 
constancy  of  the  small  event-numbers  and  that  of  relative 
numbers  with  small  basic  numbers.  The  interdependence 
of  the  single  cases  may  be  due  to  the  peculiarities  of  the 
phenomenon  in  question.  If  we  consider,  for  example,  the 
accidents  due  to  a  boiler  explosion  or  a  mine  disaster,  the 


RELATIVE  NUMBERS  AND  AVERAGES  337 

individual  deaths  are  not  independent  of  each  other,  as  in 
such  catastrophes  the  lives  of  several  persons  are  simul- 
taneously destroyed.  In  such  cases  von  Bortkiewicz  speaks 
of  an  "  acute  "  solidarity  of  the  individual  cases.  A 
"  chronic  "  solidarity  of  the  individual  cases  occurs  when 
their  interdependence  proceeds  from  influences  such  as 
climatic  conditions,  which  affect  all  of  the  cases  uniformly. 
Such  solidarity  exists,  for  example,  among  the  deaths  that 
occur  in  consequence  of  an  extremely  hot  summer.  Chronic 
solidarity  of  individual  events  may  constitute  the  rule  in 
statistics.  If  we  combine  a  great  number  of  observations 
where  such  solidarity  exists  the  statistical  masses  will 
contain  many  interdependent  units  and  the  actual  disper- 
sion must  exceed  normal  dispersion  in  a  degree  dependent 
upon  the  number  of  items  bound  together,  while  a  closer 
approximation  to  normal  dispersion  wiU  follow  a  limitation 
to  smaller  masses.^"^ 

Von  Bortkiewicz  has  arrived  at  the  conclusion  that  the 
statistical  results  satisfy  the  standard  mathematical  formu- 
las better,  if  the  field  of  observation  is  smaller  or  if  the 
event  in  question  is  very  unusual  in  a  given  society  (such 
as  suicide  or  accident )."''  The  examples  which  von  Bort- 
kiewicz used  to  support  his  conclusion  are  not  numerous, 
but  he  anticipates  that  others  will  be  found  and  that  they 
will  help  to  confirm  the  scientific  conviction  "  that  math- 
ematical probabilities  or  their  functions  underlie  all  num- 
bers relating  to  population  or  moral-statistical  phenomena." 
In  this  sense  the  law  of  small  numbers  appears  to  be  a 
suitable  "  support  for  the  explanation,  in  which  statistical 
numbers  are  consequences  of  certain  general  conditions,  to 
which  accidental  causes  are  added,  and  in  this  way  new 
authority  may  be  given  to  that  theory  of  a  specific  statistical 

*'"  Cf.  V.  Bortkiewicz,  Gesetz,  Anlage  2,  and  Tschuprow,  "  Die 
Aufgaben  der  Theorie  der  Statistik,"  Schmoller's  Jahrbuch,  1905, 
p.  57. 

i"' Gesetz,  p.  36. 


338    DISPERSION  ABOUT  THE  MEAN  OR  AVERAGE 

regularity,  which  has  been  almost  discredited  by  the  blun- 
ders of  Quetelet  and  his  followers."  ^°^ 

The  remarkable  permanence  of  certain  small  numbers  of 
events  was  proven  to  be  well-founded  by  Bowley,  inde- 
pendently of  von  Bortkiewicz 's  Das  Gesetz  der  kleirien 
Zahlen.  Bowley  cites  as  illustration  the  number  of  deaths 
in  Great  Britain  from  splenic  fever.^''^  These  numbers 
for  the  period  1875-1894  were  5,  4,  10,  14,  12,  18,  9,  15,  8, 
18,  11,  11,  11,  12,  7,  4,  3,  6,  7,  10.  The  average  is  10; 
which  is  extremely  minute  in  comparison  with  an  annual 
average  number  of  deaths  amounting  to  530,000,  and  which 
corresponds  to  an  extraordinarily  small  probability.  The 
fluctuations  of  the  number  of  deaths  from  splenic  fever  are 
consistent  with  the  theory  of  probability.  There  are  other 
small  numbers  of  events  which,  according  to  Bowley,  often 
present  extraordinary  permanence,  seldom  increasing  much 
and  just  as  seldom  entirely  disappearing.  "  Specialists  in 
all  professions,  from  the  doctor  who  treats  only  one  obscure 
disease  of  the  ear,  to  the  dealer  in  curiosities,  make  their 
livelihood  dependent  on  this  permanence  of  small  numbers. 
The  regular  occurrence  of  accidents  and  improbable  events 
in  general  furnishes  other  examples  of  the  same  sort. ' '  ^°^ 

**^  Ibid.,  preface,  p.  vi. 

'■"  Elements  of  Statistics,  Pt.  II,  Section  IV,  "The  Permanence  of 
Certain  Small  Numbers,"  2nd  ed.,  p.  301  f. 
^»»  Ibid.  p.  302. 


i 


APPENDICES 


APPENDIX  I 

A.  SERIES  WHICH  EXHIBIT  A  CHARACTERISTIC 
REGULAR  DISTRIBUTION  OF  ITEMS  IN  OTHER  WAYS 
THAN  WITH  RESPECT  TO  THE  DISPERSION  OF  THE 
ITEMS  ABOUT  THEIR  MEAN  (SERIES  OF  CHARAC- 
TERISTIC CONFORMATION) 

We  have  mentioned  ^  several  times  the  existence  of  series 
whose  items  have  no  regular  dispersion  about  the  mean 
but  which  have  some  other  characteristic  regularity  and, 
represented  graphically,  give  a  definite  characteristic  curve. 
Such  series  cannot,  evidently,  be  adequately  represented  by 
averages.  The  peculiar  characteristic  conformation  of  the 
series  must  be  set  forth.  In  order  to  accomplish  this  it 
is  necessary  to  investigate  the  succession  or  order  of  the 
numbers  and  the  relations  between  them,  that  is,  to  con- 
sider the  entire  conformation  of  the  series,  or  the  whole 
curve.  Only  in  this  way  can  the  principle  be  found  which 
lies  at  the  basis  of  the  relation  of  each  item  of  the  series 
to  all  the  others. 

It  is  our  object  to  treat  cursorily  in  the  following  of 
those  series  which  we  designate  shortly  as  "  series  of  char- 
acteristic conformation."  A  discussion  of  such  series  is 
not  inappropriate  in  a  work  on  averages,  as  the  domain 
of  averages  is,  to  a  certain  extent,  thereby  limited 
negatively.  The  problem  of  representing  and  evaluating 
such  series  is  frequently  also  positively  connected  with 
the  problem  of  averages.  Thus,  the  investigation  of  causes 
upon  the  basis  of  quantitative  series  of  characteristic  con- 
formation may  be  considered  an  extension  of  the  similar 

^  Compare  pp.  268,  293,  and  297  f. 

841 


342  APPENDIX  I 

investigation  conducted  by  comparing  averages  and  relative 
numbers.  The  discussion  of  the  investigation  of  causes  upon 
the  basis  of  such  series  will  be  supplemented  by  a  short 
explanation  of  the  methods  of  the  investigation  of  causes 
by  means  of  a  comparison  of  geographic  and  time  series. 
Finally,  the  methods  of  measuring  the  correlation  among 
several  individual  characters  will  be  indicated,  and  thus  the 
broad  outlines  of  all  the  methods  of  statistical  investigation 
of  causes  will  be  appropriately  included  in  this  book.  The 
methods  of  the  investigation  of  causes  by  means  of  the 
comparison  of  geographic  and  time  series  and  the  methods 
of  measuring  the  correlation  among  several  individual  char- 
acters are  connected  with  the  problem  of  averages  also  by 
the  fact  that  these  methods  make  extensive  use  of  averages 
for  special  auxiliary  purposes. 

Characteristic  conformations  are  frequently  found  in  time 
series  of  the  second  and  third  groups  and  also  in  quantita- 
tive series  of  the  first  and  third  groups. 

Let  us  first  consider  time  series.  A  characteristic  con- 
formation is  shown  by  **  evolutionary  "  series  which  pre- 
sent a  definite  tendency  of  development,  and  by  "  periodic  " 
series  which  present  definite  regularly  recurring  time  move- 
ments. 

Evolutionary  series  have  been  established  for  definite 
periods  of  time  in  various  statistical  fields.  Thus,  Lehr 
found  that  in  Germany  during  1841-1885  the  ratios  of  the 
number  of  births  and  the  number  of  marriages  to  the  pop- 
ulation exhibited  a  tendency  to  increase  uniformly  with 
time,  while  the  percentage  of  illegitimate  births  to  the 
total  number  of  births  showed  a  tendency  to  decrease  uni- 
formly.2  "V^ages  of  most  modern  countries  show  a  steady 
advancement,  but  the  birth  rates  of  numerous  countries 
have  steadily  decreased  in  recent  years.  An  evolutionary 
series  consisting  of  absolute  numbers  may,  however,  lose 
its  evolutionary  character  if  its  members  are  changed  to 

*  See  Handw.  d.  Staatsw.,  article  "  Gesetz "  by  Lexis. 


1 


APPENDIX  I  343 

relative  numbers;  the  quantity  of  a  commodity  imported 
may  increase  absolutely  from  year  to  year  and  yet  the  per 
capita  amount  imported  may  remain  the  same.  On  the 
other  hand,  an  evolutionary  series  of  relative  numbers  may 
change  its  character  when  the  members  become  absolute 
numbers;  thus,  in  a  growing  population  with  a  constant 
annual  number  of  births  there  is  a  continued  decrease  of 
the  birth  rate. 

The  significance  of  an  evolutionary  tendency  naturally 
increases  with  its  geographical  extent  and  the  length  of 
time  that  it  persists.  An  evolutionary  tendency  common 
to  many  countries  is  not  seldom  spoken  of  as  a  "  statistical 
law."  Thus,  G.  von  Mayr  speaks  of  a  "  law  of  the  progres- 
sive undermining  of  the  population  in  the  central  districts 
of  great  cities  "^  and  designates  the  regularity  with  which 
urban  elements  increase  and  country  elements  decrease  as 
a  social  law  of  development  of  modern  times.* 

Evolutionary  series  arouse  interest  in  proportion  to  the 
steadiness  with  which  the  tendency  of  development  is  ex- 
pressed, that  is,  in  proportion  to  the  infrequency  of  the 
fluctuations.  The  relatively  great  stability  of  certain  evo- 
lutionary series  has  frequently  led  to  their  representation 
and  characterization  by  mathematical  formulas.  As  is 
known,  it  has  been  often  asserted  (first  by  Euler  with 
reference  to  London)  that  population  increases  geomet- 
rically, that  is,  according  to  the  compound  interest  formula.^ 
Mathematical  statisticians  have  not  infrequently  used  other 
formulas  to  present  population  statistics  or  other  data  as 
functions  of  time. 

But  the  agreement  of  the  growth  of  population  or  other 
phenomena  with   a  mathematical   function   is,   obviously, 

*  Bevolkerungsstatistik,  p.  63. 

*  Ibid.  p.  61. 

°  Malthus  draws  a  contrast  between  the  "  geometric "  measure  of 
the  population  and  the  alleged  "  arithmetic  "  increase  of  the  means 
of  subsistence. 


344  APPENDIX  I 

merely  accidental  and  is,  as  a  rule,  limited  to  short  periods 
of  time.  Even  if  the  tendency  of  development  remains 
the  same,  frequent  fluctuations  of  the  rapidity  of  movement 
appear.  Thus,  it  is  never  safe  to  assume  that  population 
will  change  in  the  same  manner  in  the  future  as  it  did  in 
the  past.  Therefore,  the  computations  of  the  length  of  time 
during  which  a  population  could  double  itself,  formerly 
common,  have  long  been  rightfully  discarded.  Nevertheless, 
it  is  necessary,  in  order  to  compute  the  number  of  popula- 
tion for  an  intra-censal  year  or  for  a  post-censal  year, 
to  assume  that  the  population  changes  regularly  and  to 
ascribe  a  definite  law  of  development  to  it.  It  is  chiefly 
for  this  purpose  that  mathematical  statisticians  have  rep- 
resented population  as  a  function  of  time.^^ 

Periodic  series  as  well  as  evolutionary  series  exhibit  a 
characteristic  conformation  and  can,  therefore,  not  be  ade- 
quately expressed  by  averages.  The  entire  conformation 
must  be  investigated  in  order  to  reveal  the  periods  of  the 
series,  the  time  when  they  appear,  and  their  intensity. 

The  phenomena  which  exhibit  certain  regularly  recurring 
periods  are  very  numerous.  The  periods  may  be  of  various 
kinds,  but  seasonal  fluctuations  are  most  common.  Most 
demographic  and  moral-statistical  phenomena  (mortality, 
births  and  marriages,  crimes,  suicide,  etc.)  all  show  the  in- 
fluence of  the  seasons  more  or  less  decidedly.  This  in- 
fluence naturally  varies  with  climatic  conditions.  A  de- 
tailed investigation  will  also  develop  the  facts  that  this 
seasonal  influence  does  not  affect  all  population  groups  uni- 
formly, such  as  age  classes,  and  that  various  causes  of 
death,  various  diseases,  various  crimes,  etc.,  possess  seasonal 
curves  peculiar  to  themselves.  Likewise,  various  economic 
phenomena  such  as  unemployment,  consumption  of  various 

*»  An  illustration  of  the  use  of  the  compound  interest  law  to 
represent  the  "  growth  element "  of  certain  financial  statistics  may 
be  found  in  J.  P.  Norton's  Statistical  Studies  in  the  New  York 
Money  Market. — Tbanslatob. 


APPENDIX  I  345 

articles,  etc.,  are  affected  by  seasonal  fluctuations.  In  Aus- 
tria the  number  of  members  of  sick-benefit  societies  (de- 
pending on  the  intensity  of  employment)  gives  the  same 
characteristic  seasonal  wave  year  in  and  year  out,  the 
trough  of  the  wave  coming  in  January  and  the  crest  in 
July  or  August. 

Aside  from  seasonal  periods  there  are  also  periods  of 
longer  duration  in  many  fields  of  statistics.  The  regular 
succession  of  periods  of  economic  expansion  and  depression 
is  well  known.  This  phenomenon  is  one  which  has  inter- 
ested many  "  crisis  theorists,"  especially  the  statisticians 
Jevons  and  Juglar.  Juglar  has  found  periodic  fluctuations 
among  births  and  marriages  of  various  countries  that  cor- 
respond to  the  economic  fluctuations.®  Other  periodic  move- 
ments are  caused  by  the  influence  of  certain  days  of  the 
week  and  hours  of  the  day  upon  certain  phenomena.'^ 

Some  phenomena  are  simultaneously  affected  by  several 
wave  movements  of  various  lengths;  thus,  many  economic 
phenomena  possess  both  seasonal  fluctuations  within  the 
year  and  periods  of  expansion  and  depression  covering 
several  years.  At  the  same  time,  many  periodic  phenomena 
are  also  subject  to  a  definite  evolutionary  tendency.  In  addi- 
tion, other  disturbances  are  frequently  caused  by  events 
which  happen  from  time  to  time,  such  as  wars,  failure  of 
crops,  epidemics,  etc.  These  various  fluctuations  may  in- 
termingle and  hide  each  other.  To  establish  the  periodicity 
of  a  series  and  the  length  of  its  periods  is,  therefore,  often 
a  very  difficult  problem  and  its  solution  necessitates  a  thor- 
oughgoing study  of  the  series. 

Series  of  quantitative  individual  observations  and  quanti- 

*  Compare  "  Y  a-t-il  des  periodes  pour  les  mariages  et  les  naissancea 
comme  pour  les  crises  commerciales  ? "  Bulletin  de  I'lnst.  int.  de 
Stat.,  Vol.  XIII,  Pt  IV,  p.  8  f. 

'  Compare,  for  example,  Enrico  Raseri,  "  Les  naissances  et  les 
deces  suivant  les  heurs  de  la  journee,"  Bulletin  de  I'lnst.  int.  de  Stat., 
Vol.  XI,  Pt.  I. 


346  APPENDIX  I 

tative  series  of  relative  numbers  or  averages  sometimes  have 
a  characteristic  conformation.  Of  the  first  group  of  series 
those  which  present  the  structure  of  a  population  according 
to  age  frequently  exhibit  a  characteristic  conformation. 
G.  von  Mayr  ^  distinguishes  various  characteristic  forms 
of  age  structure  when  series  of  ages  are  represented  graph- 
ically, such  as  the  triangular  structure  of  the  population  of 
Germany  or  the  United  States,^'^  the  bell-like  structure  of 
the  French,  the  onion-like  structure  of  the  population  of 
great  cities  and  industrial  districts,  and  the  spindle-formed 
structure  of  agricultural  districts  from  which  there  is  emi- 
gration to  cities  and  industrial  centers.  Likewise,  the  age 
composition  of  those  living  according  to  the  tables  giving 
the  number  of  survivors  at  various  ages  generally  has  a 
regular  configuration.  These  numbers  do  not,  indeed,  re- 
sult from  direct  observation  but  from  computation;  never- 
theless, they  are  individual  data  and  may  be  considered 
fictitious  individual  observations. 

Other  series  of  individual  data  to  be  considered  are  in- 
comes and  wages.  Both  incomes  and  wages  of  most  coun- 
tries give  rise  to  series  which,  quite  apart  from  the  group- 
ing of  the  items  about  their  average,  exhibit  a  characteristic 
regularity  of  conformation. 

Quaniiiative  series  of  characteristic  conformation  not  in- 
frequently arise  from  relative  numbers  and  averages  based 
upon  various  age  classes.  Thus,  Quetelet  found  a  regular 
curve  for  the  rate  of  criminality  according  to  age.  The 
most  important  series  of  this  kind  is  the  probability  of 
death  according  to  age,  which — at  least  between  certain 
limits — gives  a  characteristic  curve.  Average  wages  for 
different  age  classes  also  frequently  exhibit  regular  char- 
acteristic conformations. 

*  Bevolkerungsstatistik,  p.  76  f. 

*aCf.  United  States  Census  Bulletin  No.  13  for  "A  Discussion 
of  Age  Statistics,"  which  gives  the  diagram  of  ages  for  the  popula- 
tion of  the  United  States. — Tkanslatob. 


APPENDIX  I  347 

The  mathematical  statisticians  have  often  attempted  to 
represent  series  of  individual  observations  and  quantitative 
series  of  the  third  group,  which  exhibit  characteristic  con- 
formation, by  means  of  mathematical  formulas  and  analytic 
functions.'^  Thus,  the  number  of  survivors,  the  probability 
of  death,  or  the  frequency  of  sickness  may  be  expressed  as 
a  mathematical  function  of  age;  the  number  of  persons 
receiving  a  certain  wage  or  income  may  be  expressed  as  a 
function  of  the  amount  of  wages  or  income  received. 
Quetelet  developed  a  formula  in  his  Physics  of  Society 
which  gave  the  frequency  of  crime  as  a  function  of  age; 
he  also  represented  the  growth  of  men  according  to  age  by 
an  equation  of  the  third  degree. 

The  best  known  formula  for  expressing  mortality  as  a 
function  of  age  is  the  one  advanced  by  Gompertz  in  1825 
and  later  improved  by  Makeham.^**  Numerous  other 
writers,  such  as  Lambert,  Babbage,  Litrow,  L.  Moser,  Ed- 
monds, Lazarus,  Opperman,  Thiele,  Wittstein,  and  others, 
have  also  developed  formulas  which  express  either  the 
number  of  survivors  or  the  probability  of  death  as  a  func- 
tion of  age.  Most  of  these  writers  have  overrated  the  im- 
portance of  the  formulas  that  they  developed.  They 
thought  to  reveal  a  physiological  law  by  means  of  their 
formulas.  They  supposed  that  the  mortality  curve  pos- 
sessed a  definite  general  form  and  that,  consequently,  a 
mathematical  law  of  mortality  existed  to  which  the  mor- 
tality of  every  population  fitted;  the  variations  in  mor- 
tality among  different  populations  or  groups  of  persons  was 

•  "  Every  statistical  table  suggests  the  expression  of  the  thing 
whose  quantities  it  shows  in  one  column,  as  a  function  of  the  thing 
whose  quantities  it  shows  in  another."  (Prof.  Marshall  in  the 
article  "  On  the  Graphic  Method  of  Statistics "  in  the  Jubilee 
Volume  of  the  Koyal  Statistical  Society,  1885,  p.  255.) 

*°  It  is  Ix  =k  •s'^.  (g)c*,  where  x  is  the  age,  Ix  is  the  number 
living  at  this  age,  and  k,  s,  g,  and  c  are  constants.  See  Harald 
Westergaard,  Die  Lehre  von  der  Mortalitiit  und  Morbilitat,  2nd  ed., 
p.  201. 


348  APPENDIX  I 

provided  for  in  the  various  values  for  the  constants  which 
appeared  in  the  standard  mathematical  formula.^^ 

Modern  statisticians,  with  few  exceptions,  have  little  faith 
in  a  general,  uniform  law  of  mortality.  Mortality  is  evi- 
dently strongly  influenced  by  the  greatly  divergent  con- 
ditions, both  natural  and  social,  of  various  countries.  It 
has  also  been  demonstrated  that  mortality  changes  with 
time.  It  appears  impossible,  therefore,  to  find  a  formula 
for  mortality  which  will  be  valid  for  the  series  of  observa- 
tions of  all  countries  and  all  times.^^  However,  mathemat- 
ical formulas  which  satisfactorily  describe  definite  mortality 
curves  are  of  importance  for  such  purposes  as  the  com- 
putation of  mortality,  for  reducing  the  labor  of  certain 
computations  in  life  insurance,^^  for  interpolation,^*  and 
for  the  adjustment  of  mortality  tables.^^  The  part  which 
mathematical  formulas  play  in  the  above  cases  is,  however, 
quite  different  from  the  part  that  they  would  play  as  an 
expression  of  ' '  natural  law. ' '  In  these  cases  the  securing  of 
a  mathematical  formula  is  not  an  end  in  itself  but  its 
application  is  merely  a  means  toward  attaining  some  con- 
crete purpose.^^ 

Vilf redo  Pareto  's  mathematical  theory  of  the  distribution 
of  incomes  attracted  much  attention  when  it  was  published 
in  his  Cours  d'economie  politique  (in  1896)  and  in  other 

^^  V.  Bortkiewicz  on  "  Gesetz  der  Sterblichkeit "  in  Handw.  d. 
Staatsw. 

^^  Emanuel  Czuber,  Wahrscheinlichkeitsrechnung,  No.  197,  "Mor- 
tality Formulae." 

^'  See  V.  Bortkiewicz,  "Gesetz  der  Sterblichkeit." 

^*  Compare,  for  example,  Chap.  X,  "  Interpolation,"  Section  II, 
"  Algebraic  Treatment,"  in  Bowley's  Elements  of  Statistics. 

'"  Thus,  the  first  table  of  the  civil  service  of  Austria-Hungary  was 
graduated  by  Makeham's  formula. 

'"The  objections  that  have  been  enumerated  against  a  "law  of 
mortality "  also  hold  for  B.  Scratchley's  mathematical  formulation 
of  a  "  law  of  sickness "  in  which  the  frequency  of  sickness  is  a 
function  of  age.  (Compare  with  Westergaard,  Mortalitat  und  Mor- 
bilitat,  2nd  ed.,  pp.  89  and  201  f.) 


APPENDIX  I  349 

writings.  Pareto  collected  the  income  curves  of  several 
countries  and  times  and  found  a  mathematical  formula 
which,  by  the  insertion  of  appropriate  constants,  could  be 
made  to  agree  with  all  these  curves."  "  Here  exists  a  law, 
a  true  law,"  said  Foville  of  Pareto 's  formula,"*  and,  turn- 
ing the  law  against  socialistic  tenets,  he  added :  '  *  Our  pre- 
sumptuous reformers  will  no  more  succeed  in  changing  the 
natural  curve  of  incomes  than  they  can  vary  the  parabolic 
paths  of  projectiles  or  the  elliptic  orbits  of  the  planets." 
Foville  is  of  the  opinion  that  a  change  in  the  geometric 
shape  of  the  income  curve  is  not  to  be  expected,  but  this 
view,  he  holds,  does  not  eliminate  the  possibility  of  progress 
of  the  lower  classes,  since  Pareto 's  formula  contains  vari- 
able parameters  and  the  curve  may  become  less  steep  in 
the  course  of  time.  Pierre  des  Essars,  the  French  statis- 
tician, has  tested  Pareto 's  formula  with  the  income  statistics 
of  Austria  ^^  and  found  the  formula  to  hold  also  in  this 
case.^^*  It  is  to  be  emphasized  that  Pareto 's  formula  does 
not  postulate  the  law  of  chance  and  does  not  depend  upon 
the  theory  of  error. 

Lucien  March  has  brought  the  objection  to  Pareto 's 
formula  that  it  assumes  the  smallest  incomes  to  be  the 
most  numerous,^^  This  assumption  he  held  to  be  untrue, 
as  the  income  statistics  of  Saxony — which  possess  the 
peculiar  feature  of  having  no  inferior  limit — clearly  show. 


*' This  formula  is  nz=  — ;  in  which  x  signifies  the  income,  n  the 

X* 

number  of  incomes  equal  to  or  greater  than  x,  and  a  and  a  are 
constants  for  any  given  series. 

*^a  Economiste  frangais,  July  4,  1896. 

"  Compare  Journal  de  la  Soci6t6  de  Statistique  de  Paris,  1902, 
p.  222  f. 

*'aBut  see  criticisms  of  Pareto's  formula  by  W.  M.  Persons  in 
"  The  Variability  in  the  Distribution  of  Wealth  and  Income,"  Quar- 
terly Journal  of  Economics,  May,  1909,  and  by  M.  J.  S^ailles  in 
La  Repartition  des  Fortunes  en  France. — Translator. 

"  See  Journal  de  la  Soci6te  de  Statistique  de  Paris,  1902,  p.  152  f. 


350  APPENDIX  I 

March  has  himself  offered  a  general  formula  for  the  dis- 
tribution of  a  special  category  of  incomes,  namely  wages.^° 
He  based  the  formula  upon  French,  German,  and  American 
wage  data  and  recommended  (in  the  1902  session  of  the 
Societe  de  Statistique)  that  it  be  used  to  describe  the  dis- 
tribution of  all  incomes.  The  curve  which  corresponds 
to  March's  formula  shows  the  number  of  workmen  in  vari- 
ous wage  classes.  The  minimum  wage  is  not  the  most 
frequent  and  the  distribution  of  wages  about  the  modal 
or  "  normal  "  wage  is  unsymmetrical. 

The  views  of  mathematical  statisticians  are  divided  as 
to  the  value  of  these  and  other  mathematical  formulas  for 
the  representation  of  statistical  series.  "We  shall  not  enter 
into  the  various  technical  controversies,  among  which  that 
concerning  the  degree  of  coincidence  between  the  formulas 
and  the  statistical  data  is  most  important.  It  is,  however, 
of  general  significance  that  most  of  the  formulas  con- 
structed present  only  certain  greater  or  lesser  parts  of  the 
series  in  question ;  the  formulas  for  mortality  do  not  apply 
to  the  years  of  childhood ;  ^^  Pareto  's  formula  does  not  hold 
true  for  those  receiving  small  incomes. 

The  principal  question  is  independent  of  these  questions 
of  detail.  Do  such  mathematical  formulas  express  statistical 
laws  and  do  they  benefit  the  science?  Lexis  and  von 
Bortkiewicz  hold  such  formulas  to  be  of  little  significance. 
Lexis  pointed  out  that  they  merely  present  the  exterior  of 
a  mass  of  items,  and  that  only  approximately.  "  Thus  we 
may  describe  approximately  the  exterior  surface  of  a  sand 
heap  by  means  of  an  empirical  formula,  but  we  would  never 
consider  this  formula  as  the  law  which  had  controlled  the 

*°  See  Journal  de  la  Society  de  Statistique  de  Paris,  1898,  pp.  193  f. 
and  241  f. 

"^  As  the  practical  application  of  this  formula  is  chiefly  in  life 
insurance  written  principally  for  adults  this  objection  is  of  little 
significance. 


APPENDIX  I  351 

positions  to  be  taken  by  the  grains  of  sand."  ^^  Von  Bort- 
kiewicz  says  that  the  formula  derived  by  Galton  from 
Korosi's  natality  table  fails  to  advance  the  knowledge  of 
fecundity  a  single  step;  the  friends  of  mathematical  statis- 
tics have  often  pointed  out  the  uselessness  of  similar 
formulas.^^  Lexis,  von  Bortkiewicz,  and  Edgeworth  agree 
as  to  the  decided  inferiority  of  such  empirical  formulas 
to  the  formulas  of  the  theory  of  probability,  which  latter 
offer  a  rational  explanation  of  the  grouping  of  the  items 
about  their  average.  However,  we  must  remember  that 
the  series  which  statisticians  have  sought  to  represent  by 
empirical  formulas  are  the  very  ones  that  appear  im- 
possible to  explain  by  the  theory  of  error  or  of  probability 
because  of  the  irregular  dispersion  about  the  average.  At 
the  same  time,  such  series  exhibit  a  regular  conformation 
which  it  is  possible  to  characterize  by  a  mathematical 
formula.  Moreover,  the  formulas  derived  for  quantitative 
series  of  characteristic  conformation  are  to  be  considered 
mathematically  precise  laws  of  causation — as  will  be  shown 
in  the  following  chapter — and  therefore  they  express  rela- 
tions of  indubitable  scientific  importance. 

The  mathematical  statisticians  have,  as  we  have  men- 
tioned, not  only  characterized  series  relating  to  single 
countries  by  mathematical  formulas,  but  they  have  at- 
tempted to  obtain  formulas  for  certain  phenomena  that 
would  be  generally  valid.  For  instance,  they  attempted  to 
find  general  formulas  for  mortality  according  to  age  and 
for  the  distribution  of  incomes.  They  endeavored  to 
formulate  in  mathematical  terms  the  broad  outlines  ex- 
hibited by  the  respective  phenomena  in  various  countries 
and  at  various  times  and  to  express  the  peculiarities  of 
the  different  countries  and  times  by  giving  specific  values 
to  the  constants  of  the  mathematical  functions.     Such  gen- 

"  Zur    Theorie    der    Massenerscheinungen    in    der    menschlichen 
Gesellschaft,  p.  8. 
"  Jahrbuch  fiir  Nationalokonomie  und  Statistik,  1897,  I,  p.  127. 


352  APPENDIX  I 

eral  formulas  are,  however,  very  rare.  If  the  broad  out- 
lines of  certain  phenomena  are  the  same  at  various  times 
and  in  different  countries,  such  fact  can  of  course  also  be 
established  without  the  aid  of  higher  mathematics.  More- 
over, it  is  doubtful  whether  the  mathematical  standard  can 
be  considered  to  settle  the  issue.  Several  series  for  which 
general  mathematical  formulas  cannot  be  formed  may, 
nevertheless,  be  regarded  as  of  like  conformation  if  com- 
pared according  to  the  less  rigorous  standard  of  elementary 
mathematics.  The  series  may  then  permit  a  conclusion  as  to 
uniformity  of  cause,  whether  these  causes  be  biological  or 
be  rooted  in  the  uniformity  of  the  fundamental  social  in- 
stitutions of  various  countries. 

B.  INVESTIGATION  OF  CAUSES  UPON  THE  BASIS  OP 
QUANTITATIVE  SERIES  OF  CHARACTERISTIC  CON- 
FORMATION 

Every  series  of  regular  conformation  gives  a  picture  of 
the  association  of  two  variables,  but  only  in  the  case  of 
quantitative  series  can  an  efficient  causal  relationship  be 
deduced  from  examination  of  the  conformation  of  the  series. 

Time  series  present  the  aspect  of  a  phenomenon  accord- 
ing to  divisions  of  time.  If  the  conformation  is  especially 
regular  then  it  may  be  represented  by  a  mathematical 
formula,  in  which  the  phenomenon  presented  by  the  series 
becomes  a  function  of  time.  However,  no  deduction  of  a 
proper  causal  connection — for  instance,  the  existence  of  a 
sociological  relation — can  be  made. 

In  series  of  quantitative  individual  observations  the  two 
variables  are,  first,  the  element  of  observation  in  question 
and,  second,  the  number  of  items  belonging  to  each  grade 
of  the  element  measured.  Such  series  show,  for  instance, 
the  relation  between  the  amount  of  income  or  wages  and 
the  number  receiving  specified  amounts,  or  between  age 
and  the  number  in  each  age  class.  If  the  series  in  ques- 
tion presents  a  characteristic  conformation,  then  the  rela- 


APPENDIX  I  353 

tion  between  the  two  variables  concerned  may  be  definitely 
formulated,  perhaps  by  a  mathematical  formula  in  which 
the  frequency  of  the  items  becomes  a  function  of  their 
magnitude.  But  the  interrelation  characterized  in  this  way 
is  merely  a  mathematical  one;  a  proper  causal  connection 
does  not  exist  between  these  two  variables  any  more  than 
it  does  with  time  series.  Thus,  the  amount  of  the  income 
or  wages  received  does  not  cause  any  particular  number 
of  individuals  to  receive  such  income  or  wages,  and  age 
is  not  the  cause  of  the  unequal  frequencies  of  the  age 
classes.  The  conformation  of  such  series  is  merely  de- 
scriptive in  its  significance.  A  mathematical  function  can 
merely  give  the  contour  and  the  extent  of  the  masses  of 
items  without  discovering  or  defining  a  causal  nexus.^* 

Quantitative  series  of  the  third  group  are  quite  different 
from  time  series  and  series  of  individual  observations  in 
this  respect.  The  two  variables  concerned  are,  first,  that 
quantitative  characteristic  which  is  used  to  differentiate 
the  constituent  masses,  and  second,  the  numbers  character- 
izing those  constituent  masses,  both  of  which  are  values 
which  give  properties  of  definite  statistical  masses  and,  there- 
fore, can  stand  in  a  direct  causal  relationship  to  each  other. 
If  a  quantitative  series — or  a  considerable  part  of  one — 
exhibits  a  regular  conformation  it  signifies  that  some  re- 
lation of  dependence  exists  between  the  two  variables. 
Such  regular  conformations  result  when  the  numbers  char- 
acterizing the  constituent  masses  consistently  increase  or 
decrease  as  the  mark  of  differentiation  increases.  Exam- 
ples are,  respectively:  the  increase  of  the  probability  of 
death  with  age,  and  the  decrease  of  fecundity  with  better 
economic  position.  The  hypothesis  that  such  regularity 
originates  accidentally  can  usually  be  ruled  out.     The  regu- 

^*  In  the  theory  of  error  the  number  of  the  deviations  of  items 
from  the  mean  is  expressed  as  a  function  of  the  size  of  the  de\ia- 
tions.  Here  again  a  purely  mathematical  relationship  is  created 
which  does  not  reveal  the  cause  of  the  deviations. 


354  APPENDIX  I 

larity  of  the  conformation  indicates  a  definite  causal  law. 
In  the  examples  cited  above  the  conformation  shows  a 
definite  influence  of  age  upon  the  probability  of  death,  and 
of  economic  condition  upon  fecundity. 

Series  of  such  regularity  that  the  variables  change  in  a 
definite  ratio  (such  as  that  defined  by  the  equation  y  =  ax 
-\-  b)  are  extremely  rare.  It  is  only  by  way  of  exception 
that  cause  and  effect  can  be  brought  into  direct  relation- 
ship and  there  are  usually  causes,  other  than  the  one 
''eferred  to  in  the  series,  which  disturb  the  regularity.  But 
if  it  is  possible  to  etablish  that  one  variable  varies  directly 
or  inversely  with  the  other,  this  is  of  much  significance  and 
justifies  a  conclusion.  Sometimes  it  is  possible  to  divide 
a  series  into  several  parts,  in  each  of  which  a  peculiar 
relationship  between  the  variables  may  exist.  The  direct 
parallelisms  may  become  inverse.  Thus,  in  childhood  the 
probability  of  death  decreases  as  age  increases,  while  in 
later  years  it  increases  with  age,  and  wages  increase  with 
age  up  to  the  time  of  maximum  efficiency  and  then  de- 
crease.^^'^^* 

*"  Austrian  data  provide  an  interesting  illustration  of  the  con- 
nection beween  age  and  wages.  The  information  collected  concerning 
the  conditions  of  miners  in  the  Ostrau-Karwin  coal  district  shows  that 
miners  between  the  ages  of  36  and  40  years  receive  the  highest 
incomes.  Previous  to  that  age  group  incomes  increase  with  age 
(although  not  uniformly)  and  decrease  thereafter.  The  decrease 
from  the  highest  paid  age  class  is,  however,  very  gradual  and  less 
in  amount  than  the  increase  from  the  youngest  age  classes.  (See 
Arbeiterverhaltnisse  im  Ostrau-Karwiner  Steinkohlenreviere,  pub- 
lished by  k.  k.  Arbeitsstatistisches  Amt  im  Handelsministerium, 
Pt.  I,  p.  56,  and  graphic  representation  of  Table  VIII,  following 
p.  38.) 

A  similar  result  is  given  by  the  wage  statistics-  of  industrial 
workers  of  northern  Bohemia.  "  The  wages  of  men  increase  until 
the  age  class  31-35  years  is  reached,  remain  stationary  in  the  suc- 
ceeding age  classes  to  45  years,  and  then  decrease  with  increasing 
ages,  in  spite  of  the  advancement  of  a  number  of  the  male  workmen 
into  the  better  paid  positions   of   foreman,   inspector,   and  master 


APPENDIX  I  355 

The  relations  between  the  pairs  of  variables  may  be  very- 
diverse,  and  their  formulation  may  introduce  various  math- 
ematical functions.  Mathematical  formulas  which  represent 
the  conformation  of  quantitative  series  are,  by  their  nature, 
mathematically  precise  statements  of  cause.  Consequently, 
such  formulas  possess  more  scientific  value  than  those  which 
indicate  the  conformation  of  time  series  or  series  of  in- 
dividual observations  without  revealing  any  causal  nexus. 
To  be  sure,  the  exact  mathematical  formulation  of  com- 
plicated functional  relationships  possesses  no  great  value 
for  sociologj'-,  especially  if  no  further  explanation  can  be 
given  of  the  type  of  connection  thus  revealed. 

The  investigation  of  causes  on  the  basis  of  quantitative 
series  of  characteristic  conformation  is  subject  to  the  same 

■workmen.  Accordingly,  the  age  of  maximum  earnings  of  workmen  ia 
from  31  to  45  years."  "  Among  the  women  workers  the  great- 
est efficiency  and,  therefore,  the  highest  wage  comes  somewhat 
earlier  than  for  men,  i.  e.,  between  the  ages  25  and  35  years,  and 
for  time  wages,  peculiarly  enough,  in  the  still  earlier  age  class, 
21  to  25  years.  The  wages  of  women  do  not  rise  and  fall  with 
age  as  closely  as  do  those  of  men."  (Nordbohmische  Arbeitersta- 
tistik,  Ergebnisse  der  von  der  Reichenberger  Handels-  und  Gewerbe- 
kammer  am  1.  Dezbr.  1888  durchgefiihrten  Erhebung,  Erliluterungen 
zu  den  Tabellen,  p.  xxxvii. ) 

F.  W.  Lawrence  has  made  some  interesting  investigations  as  to 
the  connection  between  wages  and  the  size  of  the  city  in  which  the 
workmen  reside.  He  has  showTi  from  wage  data  of  the  building  and 
printing  trades  and  iron  manufacture  that,  in  general,  wages  increase 
with  the  size  of  the  city  because,  indeed,  the  "  demands  for  social 
life "  increase  with  the  size  of  the  city  where  the  workman  lives. 
(See  Local  Variations  in  Wages,  London,  1899.) 

"a  Volumes  I  to  V,  inclusive,  of  the  United  States  Bureau  of 
Labor  Report  on  Condition  of  Women  and  Child  Wage-Earners  in 
the  United  States  give  wage  data  tabulated  according  to  age  of 
the  worker.  For  instance,  in  the  incandescent  electric  lamp  estab- 
lishments, employing  2,430  women,  there  is  a  rapid  rise  of  wages 
from  age  16  to  age  20,  a  gradual  rise  from  age  20  to  age  24,  and, 
finally,  a  fall  from  age  24  to  the  group,  45  and  over,  when  the  17- 
year-old  wage  level  is  nearly  reached, — Tbanslaiob. 


356  APPENDIX  I 

restrictions  and  involves  the  same  presuppositions  as  the 
investigation  of  causes  by  comparison  of  two  averages  or 
relative  numbers.^^  A  primary  consideration  is  that  the 
parallel  or  opposite  changes  of  the  two  variables  merely 
signify  a  connection  between  them,  without  specifying 
which  is  the  cause  and  which  is  the  effect.  This  question 
may  be  a  difficult  one  and  must  be  settled  by  some  non- 
statistical  method.  Frequently,  the  causal  connection  be- 
tween the  two  phenomena  is  not  one-sided  but  mutual. 
Under  certain  conditions  the  two  phenomena  may  not  be 
in  direct  causal  relationship  but  the  correspondence  which 
they  exhibit  may  be  due  to  their  dependence  upon  a  com- 
mon, and,  perhaps,  unknown  cause. 

The  method  of  investigation  of  causes  upon  the  basis  of 
quantitative  series,  like  the  method  of  comparison  of  two 
averages  or  relative  numbers,  postulates  ceteris  paribus. 
If  the  constituent  parts  which  form  the  series  are  distin- 
guished also  by  some  characteristic  other  than  the  criterion 
used  to  differentiate  them,  then  it  is  impossible  to  specify 
the  cause  of  the  differences  of  the  magnitudes  characterizing 
such  constituents.  For  example,  if  the  mortality  of  chil- 
dren was  shown  to  increase  with  the  number  of  children 
per  family,  but  if  the  larger  families  were  also  the  poorer 
ones,  then  the  greater  mortality  might  be  due  to  greater 
poverty  as  well  as  to  larger  families. 

With  reference  to  the  ceteris  paribus  hypothesis  the 
method  of  investigation  of  causes  on  the  basis  of  quantita- 
tive series  is  superior  to  that  of  the  comparison  of  two 
averages  or  relative  numbers.  If  but  two  values  are  com- 
pared (for  instance,  mortality  in  two  occupations),  then, 
in  addition  to  the  criterion  used  in  distinguishing  the  two 
masses,  there  may  be  an  indefinite  number  of  other  differ- 
ences between  the  two  masses  which  might  cause  the  differ- 
ence noted  between  the  values  compared.     However,  if  a 

'°  Compare  with  the  chapter  on  "  Investigation  of  Causes  by  Com- 
parison of  Averages  and  Relative  Numbers,"  p.  110. 


APPENDIX  I  357 

quantitative  series  is  of  regular  conformation,  then  only 
those  additional  causes  which  act  in  a  similar  regular  man- 
ner can  complicate  the  problem.  Consequently,  it  appears 
that  a  large  number  of  causes  must  be  ruled  out  and  a 
conclusion  in  regard  to  causation  is  made  considerably 
easier. 

For  example,  suppose  we  are  investigating  the  mortality 
of  two  groups  of  the  population  which  are  distinguished 
both  by  difference  of  economic  condition  and  of  occupa- 
tion. Suppose  the  mortality  rates  are  different.  It  is 
impossible  to  ascribe  a  precise  cause  for  this  difference,  as 
it  may  be  either  economic  condition  or  occupation.  How- 
ever, if  we  investigate  the  mortality  of  a  series  of  groups 
defined  by  economic  condition  and  obtain  a  regular  curve, 
then  we  may  be  able  to  conclude  that  economic  condition 
is  the  cause  of  the  differences  in  mortality,  even  if  each 
group  according  to  economic  condition  contains  different 
occupations.  That  the  regularly  graded  mortality  is  due 
to  the  different  occupations  appears  extremely  improbable. 
The  regular  shape  of  the  curve  could  be  due  to  occupation 
only  in  case  the  occupations  belonging  to  the  different  eco- 
nomic groups  are,  by  chance,  arranged  according  to  the 
degree  of  mortality.  This  is  extremely  improbable ;  if  nec- 
essary, light  may  be  obtained  by  special  investigations  of 
another  kind. 

Only  when  there  is  question  of  a  second  cause  exhibiting 
the  same  quantitative  gradations  as  those  resulting  from 
the  criterion  applied  is  there  difficulty  in  arguing  the 
presence  of  a  causal  nexus  from  the  existence  of  a  regular 
curve.  Thus,  in  the  illustration  cited  above  of  the  mor- 
tality of  children  varying  with  the  size  of  the  family,  and 
the  size  of  the  family  varying  with  economic  condition,  the 
greater  mortality  rates  may  be  due  either  to  the  larger 
families  or  to  greater  poverty.  Frequently,  however,  there 
is  no  question  of  a  second  cause  and  the  existence  of  a 
regular  quantitative  series  indicates  that  the  regularity  is 


358  APPENDIX  I 

due  to  the  criterion  at  the  basis  of  the  grouping  and  not 
to  some  unknown  additional  cause.  Other  peculiarities 
of  the  constituent  masses  may,  indeed,  be  present.  But 
these  express  themselves,  as  a  rule,  only  through  disturb- 
ances of  the  regular  form  of  the  curve. 

The  method  of  investigation  of  causes  upon  the  basis 
of  quantitative  series  of  regular  conformation  is,  in  a  way, 
an  extension  of  the  method  of  comparison  of  single  aver- 
ages or  relative  numbers.  Conclusions  which  are  drawn 
from  the  comparison  of  two  values  may  be  controlled, 
further  developed,  and  corrected  by  the  formation  of  a 
whole  series  of  values.  Thus,  the  difference  in  mortality 
between  city  and  country  is  well  kno\\Ti.  But  is  it  possible 
to  maintain  that  mortality  changes  constantly  with  the 
size  of  the  population  group?  Ballod  has  shown  that  in 
Prussia  mortality  according  to  age  classes  is  most  un- 
favorable in  the  middle-sized  cities,  then  follow  the  large 
cities,  the  small  towns,  and,  finally,  the  country.^^  The  size 
of  the  population  group  appears  to  be  of  influence,  but 
there  is  no  consistent  parallelism.  Another  illustration 
follows. 

Some  statisticians  have  maintained  that  the  difference 
of  ages  of  parents  has  an  influence  upon  the  sex  of  the 
offspring.  Korosi  has  tested  this  question  by  observations 
in  Budapest,  with  the  result  that  in  those  eases  where  the 
fathers  were  decidedly  older  or  younger  than  the  mothers 
the  percentage  of  boys  born  was  considerably  greater,  but 
the  sex-ratio  of  the  children  did  not  vary  uniformly  with 
the  difference  of  ages  of  the  parents.^* 

"  Bulletin  de  I'lnstitut  intern,  de  Statistique,  Vol.  XIV,  No.  1, 
p.  135.  See  also  Carl  Ballod,  Die  mittlere  Lebensdauer  in  Stadt 
und  Land.  (Staats-  und  sozialwissenschaftliche  Forschungen,  edited 
by  Prof.  Schmoller,  Vol.  XVI,  Pt.  V,  1894.) 

'"  Neue  Beitrage  zur  Sexualproportion  der  Geburten.  (Bulletin 
de  I'lnstitut  intern,  de  Statistique,  Vol.  XIV,  No.  4,  p.  14.) 


APPENDIX  I  359 

C.     INVESTIGATION     OF     CAUSES     THROUGH     COM- 
PARISON OF  GEOGRAPHIC  AND  TIME  SERIES 

Investigation  of  causes  upon  the  basis  of  quantitative 
series  presupposes  the  formation  of  graded  constituent  parts 
defined  by  a  quantitative  criterion.  Many  times,  however, 
there  are  insurmountable  obstacles  to  the  prosecution  of  a 
concrete  investigation  by  the  method  described  in  the  pre- 
ceding section.  In  such  cases  we  may  have  recourse  to 
an  "  indirect  "  method.  Instead  of  directly  applying  the 
quantitative  criterion  in  question  to  form  the  constituent 
parts  of  a  series,  we  may  make  use  of  available  geographic 
or  time  masses,  if  these  masses  vary  at  the  same  time  with 
respect  to  the  quantitative  element  in  which  we  are  inter- 
ested. For  example,  suppose  that  we  are  investigating  the 
connection  between  economic  well-being  and  mortality.  If 
there  are  statistical  difficulties  which  prevent  a  classification 
of  both  the  population  and  deaths  according  to  economic 
well-being,  then  we  may  examine  a  series  based  upon  geo- 
graphic divisions,  which  divisions  contain  populations  of 
various  degrees  of  well-being,  with  reference  to  their  mor- 
tality. Or,  we  may  compare  various  periods  of  time,  during 
which  economic  conditions  are  varied,  with  reference  to 
mortality.  Which  "  indirect  "  method  we  choose  is  a 
question  of  expediency.  Many  times  both  methods  are 
equally  practicable.  Geographic  and  time  series  must  also 
always  be  utilized  if  the  various  grades  of  the  element  in 
which  we  are  interested  are  not  to  be  found  at  the  same 
time  or  in  the  same  country. 

However,  it  may  happen  that  geographic  or  time  series 
do  not  give  the  solution  of  quite  the  same  problem  to 
which  the  comparison  of  quantitative  constituents  of  the 
same  totality  relates.  In  such  cases  the  indirect  method 
possesses  independent  significance,  but  of  course  it  cannot 
be  used  as  a  substitute  for  the  direct  method.  Thus  the 
influence  of  economic  well-being  will  be  found  to  be  essen- 
tially different  in  case  w^e  study  the  various  degrees  of  well- 


360  APPENDIX  I 

being  which  exist  permanently  for  various  social  classes 
as  compared  with  those  which  result  from  quickly  alternat- 
ing periods  of  industrial  depression  and  expansion  affecting 
the  whole  population. 

If  we  depend  upon  a  geographic  series  to  establish  a 
certain  causal  relationship  we  have  to  arrange  the  geo- 
graphic divisions  of  the  series  according  to  the  magnitude 
of  both  quantitative  elements  between  which  the  de- 
pendence is  supposed  to  exist.  Thus,  we  could  easily 
arrange  the  series  according  to  mortality  and  according 
to  economic  well-being.  If  the  two  characteristics  are  really 
causally  connected,  then  the  geographic  divisions  in  ques- 
tion will,  on  the  whole,  appear  either  in  the  same  order  or 
in  opposite  order  in  both  arrangements  according  as  the  re- 
lation is  direct  or  inverse. 

Such  comparisons  of  geographic  series  are  quite  frequent 
and  numerous  causal  relationships  are  thus  ascertained. 
The  comparison  of  series  fulfils  its  purpose  as  completely, 
however,  when  an  expected  relationship  is  contradicted  as 
when  a  causal  nexus  is  demonstrated.  Comparisons  of  geo- 
graphic series  have  been  made  to  ascertain  the  connection 
between  economic  well-being  and  mortality,  well-being  and 
number  of  children,  birth  rates  and  mortality  of  children, 
size  of  farms  and  density  of  population,  mortality  and 
density  of  population,  etc.  It  has  been  repeatedly  estab- 
lished in  England  that  the  most  densely  populated  regis- 
tration districts  also  have  the  highest  mortality  rate.  The 
earlier  English  statisticians  saw  in  this  parallelism  of  in- 
creasing density  of  population  and  rate  of  mortality  the 
expression  of  a  general  statistical  law.  However,  this 
parallelism  is  not  exhibited  by  the  more  recent  statistics  of 
other  countries.-® 

Geographic  series  based  upon  density  of  population  and 
upon  birth  rates  have  also  been  compared.  The  question 
whether  the  marriage  rate  and  the  frequency  of  illegitimate 

^*  Compare  G.  v.  Mayr,  Bevolkerungsstatistik,  p.  222  f. 


APPENDIX  I  3G1 

births  are  connected  has  been  advanced  by  J.  Bertillon. 
A  priori  one  might  expect  that  where  the  marriage  rate 
is  higher  fewer  illegitimate  children  would  be  born.  A 
comparison  of  geographic  series  has  not  confirmed  this 
expectation.^"  J.  Bertillon  has  likewise  investigated  the 
question  of  the  relation  between  the  frequency  of  illegiti- 
mate births  and  age  of  marriage.  The  comparison  of 
such  series  in  fact  indicated  a  connection  between  a  higher 
age  at  marriage  and  a  greater  frequency  of  illegitimate 
births,  and  conversely. 

Numerous  other  illustrations  of  the  application  of  these 
methods  might  be  given.  Among  these  there  are  some  cases 
which  have  been  interpreted  variously,  thus  leading  to  con- 
troversies. Thus,  Achille  Guillard's  "  Loi  du  Rapport 
inverse,"  according  to  which  growth  of  population  stands 
in  inverse  relation  to  density  of  population,  caused  much 
dispute.  In  his  Elements  de  Statistique  Humaine,  ou 
Demographie  Comparee,  which  appeared  in  1855,  Guil- 
lard  arranged  114  countries  and  provinces,  first  according 
to  their  respective  densities  of  population  and,  second, 
according  to  the  rate  of  growth  of  their  population,  and 
he  found  that,  on  the  whole,  the  geographic  divisions  ap- 
peared in  opposite  order.  The  violence  with  which  Wap- 
paus,  for  example,  has  opposed  this  law  is  interesting.^^ 
Only  one  of  Wappaus's  numerous  arguments  in  opposition 
will  be  cited.  He  contended  that  it  was  fallacious  for 
Guillard  to  base  his  conclusions  upon  averages  for  great 
geographic  areas  consisting  of  heterogeneous  parts,  such  as 
the  density  of  population  of  Russia.  Such  averages,  which 
may  differ  widely  from  the  individual  conditions  which 
they  represent,  are  quite  frequently  used  in  comparisons 
of  geographic  series  (for  instance,  when  nations  are  used  as 
units),  and  consequently  the  conclusions  are  often  question- 

'"  Compare  J.  Bertillon,   Cours  ^Ifimentaire  de  Statistique  admi* 
nistrative,  p.  480. 

*^  AUgemeine  Bevolkerungsstatistik,  Pt.  I,  p.  144  f. 


362  APPENDIX  I 

able.  This  objection  to  the  method  of  comparison  of  geo- 
graphic series  can,  however,  be  overcome  by  using  ' '  natural 
districts,"  when  possible,  instead  of  political  divisions. 
Detailed  investigations  may  enable  us  to  form  homogeneous 
geographic  complexes,  of  a  greater  or  smaller  size,  which 
correspond  to  the  various  gradations  of  a  definite  quantita- 
tive characteristic  (for  example,  natural  areas  for  various 
degrees  of  mortality).  These  statistical  districts  which 
correspond  to  the  various  degrees  of  a  definite  phenomenon 
may  then  be  examined  with  reference  to  any  other  phenom- 
enon in  which  we  are  interested  to  see  if  the  two  phenomena 
are  related.  If  the  second  phenomenon  exhibits  the  same — 
or  the  inverse — arrangement  of  areas,  when  the  items  of 
the  two  series  are  placed  according  to  magnitude,  it  means 
that  the  two  phenomena  are  causally  connected.  G.  von 
Mayr  was  able  to  form  well  defined  districts  in  Bavaria  dis- 
tinguished by  various  degrees  of  child  mortality.  He  then 
ascertained  the  density  of  population  for  these  areas,  and 
the  birth  rate  and  the  frequency  of  illegitimate  and  still- 
births as  well,  and  in  this  way  investigated  the  relationship 
between  these  phenomena. ^^  In  a  similar  manner  we  could 
form  "  natural  districts  "  according  to  density  of  popula- 
tion or  size  of  farms  and  ascertain  the  amount  of  emigration 
from  these  divisions  to  cities  or  to  foreign  countries,  or  the 
degrees  of  any  other  phenomenon  which  might  be  depend- 
ent upon  density  of  population  or  the  distribution  of  land 
ownership. 

Time  series  are  compared  just  as  frequently  as  are  geo- 
graphic series  in  order  to  determine  whether  their  con- 
formations are  similar  and  whether  a  causal  connection  can, 
therefore,  be  assumed.  The  coincidence  of  two  time  series 
may  be  either  in  their  evolutionary  tendency  or  in  their 
concomitant  or  synchronous  oscillations.  Whenever  a 
causal  nexus  is  assumed,  whether  the  two  series  show  paral- 
lel or  opposite  variation,  we  speak  of  the  items  as  corre- 

*'  Compare  G.  v.  Mayr,  Theoretische  Statistik,  p.  88. 


APPENDIX  I  363 

lated  quantities.  The  comparison  of  time  series  is  of 
interest  even  though  there  is  no  question  of  a  causal  con- 
nection between  the  two  phenomena.  We  might  investigate 
the  question  whether  there  is  a  progress  in  the  economic 
well-being  of  a  country  corresponding  to  the  per  capita 
increase  of  taxes  (per  capita  imports  plus  exports,  or  per 
capita  consumption  being  used  as  indices  of  well-being). 
Or,  we  might  inquire  if  any  selected  phenomenon  (such 
as  foreign  trade)  exhibits  the  same  evolutionary  tendency 
in  various  countries,  etc. 

The  comparison  of  the  course  of  two  or  more  time  series 
is  greatly  facilitated  by  graphic  representation.  The  syn- 
chronous oscillations  of  "  historical  curves  "  are  much 
easier  to  locate  than  similar  variations  in  the  numerical 
data.  Graphic  representation  is,  therefore,  strongly  ad- 
vised particularly  for  "  experimental  "  investigation  of 
causes.  If  various  types  of  phenomena  (such  as  foreign 
trade,  marriage  rate,  unemployment,  etc.)  are  represented 
in  the  same  diagram,  then  the  impression  of  the  diagram 
and  its  conclusiveness,  of  course,  depend  upon  the  scales 
chosen  for  the  several  curves  and  the  relation  of  the  scales 
to  each   other. ^^"^ 

A  certain  space  of  time  must  often  elapse  between  the 
operation  of  cause  and  effect.  In  such  a  case  the  move- 
ments of  the  two  curves  are  not  synchronous  but  are  sepa- 
rated by  that  part  of  the  curve  corresponding  to  the  elapsed 
time.  Thus,  R.  H.  Hooker  ^^  has  found  that  in  England 
the  parallelism  of  marriage  rates  and  foreign  trade  during 
the  period  1861-1895  is  not  greatest  for  simultaneous  fluctu- 
ations but  for  marriage  rates  and  imports  (or  total  foreign 
trade)  which  precede  the  marriage  rates  by  a  third  of  a 

»*aCf.  Bowley,  Elements  of  Statistics,  Chap.  VII,  "The  Graphic 
Method,"  for  a  discussion  of  the  comparison  of  curves  and  the  choice 
of  scales. — Translator. 

""Correlation  of  the  Marriage-Rate  with  Trade."  (Jour,  of  the 
Royal  Stat.  Soc,  1901,  p.  487  f.) 


364  APPENDIX  I 

year,  or  exports  which  precede  marriage  rate  by  about 
half  a  year.  Hooker  found  that  a  year  and  a  quarter 
intervened  between  total  bank  clearings  and  marriage 
rates. 

The  earlier  statistical  text-books  usually  gave  the  fre- 
quently observed  correspondence  between  the  number  of 
births  and  marriages  and  the  price  of  grain  as  an  illustra- 
tion of  correlated  time  series.  The  price  of  grain  was  taken 
as  an  index  of  economic  conditions.  However,  it  is  no 
longer  an  accurate  index  of  conditions  and,  consequently, 
the  parallelism  that  formerly  held  true  has  not  existed  for 
some  decades.  Phenomena  which  have  been  more  recently 
used  as  indices  of  the  economic  conditions  and  the  move- 
ments of  which  have  been  compared  with  each  other  and 
with  births,  deaths,  marriages,  etc.,  are  the  state  of  the  labor 
market,  the  per  capita  amount  of  foreign  trade,  the  amount 
of  savings,  the  consumption  of  certain  articles,  the  percent- 
age of  poor  receiving  government  aid,  the  per  capita  amount 
of  bank  clearings  (the  last  appears  in  the  official  annual 
Report  of  the  English  Registrar  General),  etc.^^* 

The  fluctuations  of  criminality  and  the  changes  in  the 
price  of  grain  have  likewise  been  compared,  and  thus  the 
well-knowTi  parallelism  between  the  price  of  grain  and 
larceny,  and  the  inverse  relation  between  the  price  of  grain 
and  bodily  assault,  have  been  established.  But  these  rela- 
tions no  longer  exist,  probably  for  the  reason  that  the  price 
of  grain  no  longer  serves  as  a  barometer  of  the  economic 
condition  of  the  great  masses  of  the  population.  The  same 
statement  holds  true  of  the  coincidence  between  grain  prices 
and  the  number  of  German  emigrants,  which  items  moved 
together  previous  to  1870.     A  great  number  of  influences, 

**aA  number  of  series  of  economic  statistics  showing  synchronous 
fluctuations  have  been  charted  by  Mr.  W.  H.  Beveridge  in  his  Un- 
employment. The  chart  is  given  the  significant  title  "The  Pulse- 
beat  of  the  Nation." — Tbanslatob. 


APPENDIX  I  365 

which  have  nothing  to  do  with  the  price  of  grain,  now 
affect  the  number  of  emigrants.^* 

Of  more  recent  date  are  Juglar's  investigations  concern- 
ing the  occurrence  of  economic  crises  at  the  times  when 
bank  loans  were  highest  and  bank  reserves  lowest.  In  order 
to  indicate  what  a  great  variety  of  series  may  be  compared 
it  is  sufficient  to  mention  the  attempt  to  explain  the  often 
observed  simultaneous  appearance  of  sun-spots  and  eco- 
nomic crises;  the  demographic  congress  of  1887  considered 
this  question  as  well  as  that  of  the  relation  between  sun- 
spots  and  mortality.  An  English  author  has  even  discussed 
the  connection  between  English  death  rates  and  the  orbital 
motions  of  the  planet  Jupiter.^^ 

Just  as  it  is  the  tendency  of  other  methods  of  modem 
statistics  to  depend  more  and  more  upon  minute  investiga- 
tions, so  in  the  comparison  of  time  series  more  attention  is 
now  paid  to  the  details  of  the  phenomena.  It  is  especially 
practicable  to  take  into  consideration  the  differences  between 
various  social  classes.  If  we  are  investigating  a  causal 
connection  it  is  evident  that,  where  possible,  only  those 
masses  should  be  compared  which,  on  the  one  hand,  express 
cause  and,  on  the  other  hand,  express  effect.  To  include 
in  the  comparison  masses  which  do  not  participate  in  the 
causal  connection  must  evidently  spoil  the  picture  and 
make  the  demonstration  more  difficult.  G.  von  ]\Iayr  has 
emphasized  this  idea  in  connection  with  the  special  problem 
of  the  relation  between  criminality  and  the  price  of  grain. 
He  makes  the  point  that,  evidently,  "  millionaires  are 
not  immediately  driven  to  larceny  by  an  increase  in  the 
price  of  grain  ";  "it  is,  therefore,  not  sufficient  to  com- 
pare total  criminality  with  the  price  of  grain;  if  we  wish 
to  get  valuable  results  we  must  pay  especial  attention  to 

'*  Compare  G.  v.  Mayr,  Bevolkerungsstatistik,  p.  347. 

"  B.  G.  Jenkins,  "On  a  Probable  Connection  behveen  the  Yearly 
Death-rate  and  the  Position  of  the  Planet  Jupiter  in  His  Orbit." 
(Jour,  of  the  Roy.  Stat.  Soc,  1879,  p.  330.) 


366  APPENDIX  I 

the  criminality  of  various  social  classes."^®  It  is  just  as 
evident  that  the  size  of  the  crop  and  the  price  of  grain 
must  exert  different  influences  upon  the  producing  and 
the  consuming  classes,  whether  we  are  examining  the  move- 
ment of  the  population,  marriages,  or  crimes.  Nevertheless, 
this  difference  was  almost  neglected  by  the  earlier  writers. 
By  distinguishing  between  producers  and  consumers  B. 
Pokrovsky  has  obtained  valuable  new  results  in  his  study 
of  the  influence  of  crops  and  the  price  of  wheat  upon  the 
movement  of  population  in  Russia,^^  as  has  also  Dr.  J. 
Buzek,  who  investigated  the  influence  of  crops  and  prices 
of  grain  upon  the  movement  of  the  population  of  Galicia 
(a  province  of  Austria)  during  the  period  1878-1898.^^ 

If  the  similarity  of  the  fluctuations  of  two  series  is  es- 
tablished, then  we  may  proceed  to  investigate  the  ratio 
between  the  fluctuations.  For  instance,  is  the  change  in 
the  marriage  rate  always  a  certain  percentage  of  the  change 
in  the  amount  of  exports?  As  a  rule  we  must  be  satisfied 
if  we  find  that  greater  fluctuations  in  one  series  are  accom- 
panied by  greater  fluctuations  in  the  other  series,  as  exact 
proportions  seldom  exist.^^ 

If  an  exact  expression  of  the  degree  of  parallelism  be- 
tween two  series  is  desired,  then  we  may  look  for  a  numeri- 
cal measure  which  shall  vary  with  the  degree  of  similarity 
between  corresponding  fluctuations  of  all  the  items  of  the 
two  series.  Thus,  pairs  of  series  may  be  graded  according 
to  the  degree  of  parallelism.  March  and  the  English  statis- 
ticians have  paid  especial  attention  to  the  derivation  of 

*»"Uber  die  statistisehen  Gesetze."  (Bull,  de  I'Inst.  intern,  de 
Stat.,  Vol.  IX,  No.  2,  p.  309.) 

*''  Given  at  the  St.  Petersburg  session  of  the  International  Sta- 
tistical Institute  in  1897.  See  Bull,  de  I'Inst.  intern,  de  Stat.,  Vol. 
XI,  No.  1,  Pt.  II,  p.  176. 

"  Statistische  Monatsschrift  (Vienna),  1901,  pp.  167-216. 

"  Bowley  has  described  a  special  graphic  method  of  testing  the 
proportionality  of  the  fluctuations  of  two  series.  (Elements  of  Sta- 
tistics, 2nd  ed.,  p.  177.) 


APPENDIX  I  367 

such  a  numerical  measure.  The  former  has  constructed  a 
"  coefficient  de  dependance  "  upon  elementary  mathemat- 
ical formulas.  By  means  of  this  coefficient,  the  correspond- 
ence of  the  fluctuations  of  two  time  series  is  summarized 
and  an  index  is  obtained  which  varies  with  the  degree  of 
coincidence  of  the  series  compared.*"  The  English  mathe- 
matical statisticians  have  developed  a  "  coefficient  of  corre- 
lation," based  upon  the  calculus  of  probability,  which  varies 
with  the  degree  of  correlation  between  the  two  series  com- 
pared. Numerous  ingenious  mathematical  studies  of  the 
theory  of  correlation  have  been  made.*^-*^*  Whether  any 
two  phenomena  are  dependent  or  not  is  judged  by  the  mag- 
nitude of  the  coefficient  of  correlation  computed  for 
them.*^'' 

*°  See  "Comparison  num^rique  de  eourbes  statistiques."  (Jo urn.  de 
la  Soc.  de  Stat,  de  Paris,  1905,  pp.  255  f.  and  306  f.) 

"  Compare  Pt.  II,  Section  VI,  "  The  Theory  of  Correlation "  in 
Bowley's  Elements  of  Statistics.  The  most  noteworthy  writers  on 
the  subject  are  Edgeworth,  Pearson,  Galton,  Yule,  Hooker,  and 
Sheppard. 

*'a  For  a  resume  (mathematical)  of  the  work  in  this  field,  together 
with  some  new  applications,  see  "  The  Correlation  of  Economic 
Statistics  "  by  W.  M.  Persons  in  the  Quar.  Publics,  of  the  Am.  Stat. 
Assoc,  for  December,  1910.  For  an  extensive  mathematical  treat- 
ment of  frequency-curves,  dispersion  and  correlation,  see  Yule's  An 
Introduction  to  the  Theory  of  Statistics  (1911).  For  a  non-mathe- 
matical explanation  of  the  meaning  of  the  standard  deviation,  co- 
efficient of  correlation,  etc.,  see  W.  P.  and  E.  M.  Elderton's  Primer  of 
Statistics    (1909). — Translator. 

*^bThe  coefficient  of  correlation  was  first  derived  by  A.  Bravais 
in  1846,  who,  however,  did  not  use  a  separate  symbol  to  stand  for  the 
function,  nor  did  he  apply  it  to  statistics.  Galton,  Pearson,  Edge- 
worth,  and  Yule  have  applied  the  function  to  statistics  and  devel- 
oped the  theoretical  side. 

The  coefficient  of  correlation  was  derived  by  assuming  that  a  large 
number  of  independent  causes  operate  upon  each  of  two  series,  pro- 
ducing normal  distributions  in  both  eases.  Upon  the  assumption 
that  the  set  of  causes  operating  upon  the  first  series  is  not  inde- 
pendent of  the  set  of  causes  operating  upon   the  second  series  a 


368  APPENDIX  I 

Certain  authors  have  used  the  theory  of  correlation  also 
to  express  the  relationship  between  three  variables.  Social 
phenomena  are,  as  a  rule,  dependent  upon  very  many  causes. 
Instead  of  relating  a  phenomenon  to  a  single  cause — as  is 
commonly  done — we  may  seek  to  determine  its  dependence 
upon  two  or  more  causes.  Thus,  Hooker  and  Yule  have 
estimated  the  influence  of  price  and  of  the  quantity  of 

function  is   found  of  the   form    — ^—^.     This   is   the   coefficient  of 

n<ri(r2 

correlation  (r).     The  notation  is  as  follows: 

X  =  any  measurement  in  the  first  series. 

Y  =  any  measurement  in  the  second  series. 

X  rr  deviation  of  any  item  of  the  first  series  from  the  arithmetic 
mean  of  the  series. 

y  =z  deviation  of  any  item  of  the  second  series  from  the  arithmetic 
mean  of  the  series. 

ffi  z=  standard  deviation  of  the  X  series. 

<r,  =  standard  deviation  of  the  Y  series. 

n  =:  number  of  items  in  each  series. 

It  has  been  demonstrated  that  r  cannot  be  greater  than  -j-  1  nor 
less  than  —  1.  Positive  values  of  r  mean  that  large  items  of  the 
X  series  occur  simultaneously  with  (are  paired  with)  large  items  of 
the  Y  series;  negative  values  of  r  mean  that  large  values  of  the 
X  series  occur  with  small  values  of  the  Y  series.  A  value  of  r 
approximating  O  means  that  there  is  no  correlation  between  the  two 
series.  There  can  be  perfect  positive  correlation  (r=  +1)  or 
perfect  negative  correlation  ( r  ^  —  1 )  only  in  case  the  items  of 
the  two  series  are  connected  by  a  function  of  the  first  degree 
(graphically,  a  straight  line). 

I_r2 

Karl  Pearson  has  found  the  probable  error  of  r  to  be  0.67  — ?=■• 

A.  L.  Bowley  has  laid  down  the  following  rule  for  judging  the  mean- 
ing of  r :  "  When  r  is  not  greater  than  its  probable  error  we  have  no 
evidence  that  there  is  any  correlation,  for  the  observed  phenomena 
might  easily  arise  from  totally  unconnected  causes;  but  when  r  is 
greater  than,  say,  6  times  its  probable  error,  we  may  be  practically 
certain  that  the  phenomena  are  not  independent  of  each  other,  for 
the  chance  that  the  observed  results  would  be  obtained  from  un- 
connected causes  is  practically  zero."  (Elements,  2nd  ed,,  p.  320.)  — 
Tbanslatob. 


APPENDIX  I  369 

wheat  produced  upon  the  wheat  exports  from  India.*^  The 
mathematical  treatment,  of  course,  becomes  considerably 
complicated  when  more  than  two  variables  are  used. 

The  investigation  of  causes  by  means  of  the  comparison 
of  geographic  and  time  series  is  subject  to  similar  limita- 
tions as  the  investigation  of  causes  by  means  of  quantitative 
series  of  characteristic  conformation,  and  by  the  compari- 
son of  two  averages  or  relative  numbers.*^  Which  of  two 
apparently  related  phenomena  is  cause  and  which  is  effect 
cannot  be  determined  by  the  statistical  comparison.  The 
two  phenomena  may  coincide  because  they  have  a  common 
cause  or  causes,^*  or  each  may  react  upon  the  other. 
Mutual  influences  are  known  to  be  especially  common  among 
social  and  economic  phenomena.  Thus — to  give  but  two 
such  cases  that  may  be  investigated  by  means  of  time  series 
— there  is  a  reciprocal  action  between  prices  and  consump- 
tion, and  between  freight  rates  and  tonnage. 

Further,  it  is  to  be  borne  in  mind  that  conclusions  in 
regard  to  causation  based  upon  geographic  and  time  series 
are  always  hypothetical,  that  is,  they  always  rest  upon  the 
assumption  ceteris  paribus.  However,  the  investigations 
of  causation  by  the  method  of  comparison  of  series  and 
upon  the  basis  of  regular  quantitative  series  are  in  a  better 
position  as  regards  this  assumption  than  is  the  method  of 
the  comparison  of  averages  or  relative  numbers.  If  a  num- 
ber of  geographical  divisions  are  arranged  according  to 
the  intensity  of  each  of  two  criteria  and  if  the  items  of 

**  "  Note  on  Estimating  the  Relative  Influence  of  Two  Variables 
upon  a  Third."     (Journ.  of  the  Roy.  Stat.  Soc.,  1906,  p.  197.) 

"  See  above,  pp.  110  f.  and  353  f. 

**  The  parallelism  between  divorce  and  suicide  established  by  J. 
Bertillon  upon  the  basis  of  geographical  comparison  should  be  men- 
tioned in  this  connection.  The  countries  with  a  high  suicide  rate 
show,  in  general,  a  relatively  high  divorce  rate.  This  parallelism 
does  not  rest  upon  a  direct  causal  connection,  but  may,  however, 
result  from  common  causes.  Dipsomania,  for  example,  leads  to  the 
increase  both  of  suicide  and  of  divorce. 


370  APPENDIX  I 

both  series  appear  in  the  same  order,  then,  as  a  rule,  we  may- 
assume  the  existence  of  a  causal  connection  between  them, 
since  it  is  extremely  improbable  that  the  selfsame  arrange- 
ment occurs  in  both  by  chance.  Also,  in  the  case  of  the 
comparison  of  time  series  which  vary  together  and  exhibit 
corresponding  fluctuations  it  is  difficult  to  find  any  valid 
explanation  other  than  that  of  a  direct  or  indirect  causal 
connection  between  two  phenomena.  That  two  independent 
phenomena  should  give  rise  to  similar  curves  with  syn- 
chronous fluctuations  during  any  considerable  time  would 
be  extraordinarily  improbable. 

D.    CORRELATION  BETWEEN  INDIVIDUAL  CHAR- 
ACTERS 

The  investigation  of  the  association  of  two  individual 
characters  (elements  of  observation,  individual  measure- 
ments) with  reference  to  the  correlation  existing  between 
them  is  essentially  allied  to  the  investigation  of  causation 
by  comparison  of  two  time  series.*^  Two  individual  char- 
acters are  said  to  be  correlated  when  the  variations  of  one 
character  are,  on  the  whole,  matched  by  corresponding  varia- 
tions of  the  other  character.  In  such  a  case  a  direct  causal 
connection  may  be  present,  so  that  the  changes  in  one 
variable  may  be  the  cause  of  the  changes  in  the  other;  but 
there  may  be  a  common  cause  producing  the  fluctuations  in 
both  characters.  It  is  possible  to  investigate  the  correlation 
between  two  characters  only  in  case  they  belong  to  the  same 
individuals  observed,  or  if  the  measurements  of  one  phe- 
nomenon may  be  paired  with  measurements  of  another 
phenomenon.  The  second  case  is  similar  to  the  case  of 
two  time  series  in  which  every  item  of  the  first  series  is 
paired  with  an  item  of  the  second  series  relating  to  the  same 
period  of  time  (year,  month,  etc.). 

"  Such  data  of  observation  are  presented  in  a  "correlation  table," 
which  is  a  double  frequency  table,  i.  e.,  each  element  appears  simul- 
taneously in  two  classes  of  measurement. 


APPENDIX  I  371 

Measurements  of  two  or  more  characters  of  the  same  in- 
dividuals are  frequently  utilized  for  the  measurement  of 
correlation  in  biology  and  anthropology.  For  example,  if 
the  lengths  of  two  different  parts  of  the  body  are  measured 
for  a  group  of  individuals  we  may  determine  whether,  as 
a  rule,  those  individuals  which  possess  a  larger  character 
of  the  first  kind  also  possess  a  larger  character  of  the  second 
kind.  If  this  is  the  case  the  two  characters  measured  are 
positively  correlated.  As  a  matter  of  fact,  research  has 
shown  that  biological  and  anthropological  characters  are 
usually  correlated  to  a  sufficient  degree,  so  that  changes  in 
one  character  are  accompanied  by  definite  positive  or  nega- 
tive change  in  other  characters.*®  The  greatest  degree  of 
correlation  is  exhibited  by  the  right  and  left  parts  of  ani- 
mals. Thus,  Galton  has  applied  the  method  of  measuring 
correlation  to  the  numbers  of  Miillerian  glands  on  the  right 
and  left  sides  of  swine.  The  number  of  these  glands  varies 
widely  with  the  individual.  In  case  of  absolute  symmetry 
there  would  be  the  same  number  of  glands  on  both  right 
and  left  sides  and  the  correlation  would  be  perfect.  As  a 
matter  of  fact,  although  the  correlation  is  not  perfect  it 
is  very  great. 

A  table  giving  the  ages  of  bride  and  groom  at  marriage 
offers  an  illustration  of  combined  observations  not  belong- 
ing to  the  same  individuals  but  still  suitable  for  the 
application  of  the  test  for  correlation.  The  two  variables 
are  the  ages  at  marriage  of  the  two  sexes.  The  problem 
may  be  stated  as  follows:  does  the  age  of  the  groom  at 
marriage  vary  in  a  definite  way  with  the  age  of  the  bride 
at  marriage  ?  ^"^ 

Much  light  has  been  thrown  upon  the  problems  of  hered- 

*'  Compare  Georg  Duneker,  Die  Methode  der  Variationsstatlstik 
(1899),  II,  "Correlation,"  III,  "Some  Problems  of  Statistical 
Method." 

*'  Compare  G.  Udny  Yule,  on  the  "  Theory  of  Correlation."  ( Journ. 
of  the  Roy.  Stat.  Soc,  Vol.  LX  (1897),  p.  813.) 


372  APPENDIX  I 

ity  and  selection  by  recent  applications  of  the  theory  of  cor- 
relation. Pearson  and  Galton  have  been  pioneers  in  this 
field,  where  observations  belonging  to  different  individuals 
are  paired.  Galton,  for  instance,  has  compared  the  stature 
of  parents  and  children  and  found  that  the  average  stature 
of  the  sons  born  of  fathers  of  a  given  stature  is  nearer  to 
the  average  stature  of  the  population  than  is  the  stature 
of  the  fathers.  This  phenomenon  was  called  *  *  regression. ' ' 
The  coefficients  of  correlation  and  regression  afford  a 
method  of  measuring  the  force  of  heredity  and  the  effects 
of  natural  selection.'** 

The  same  mathematical  methods  which  are  being  applied 
to  the  measurement  of  correlation  between  the  items  of 
two  time  series  can  also  be  applied  to  the  association  of 
two  Individual  characters.*^"^'*  But  the  problems  which  are 
dealt  with  in  this  way  are,  by  no  means,  peculiar  to  mathe- 
matical statistics.  They  likewise  confront  the  statistician 
who  merely  uses  elementary  mathematics.  Of  course  he 
will  be  obliged  to  make  approximate  judgments  and  will 
not  be  able  to  get  such  precise  results  as  can  be  obtained 
by  the  refined  mathematical  methods. 

**  Compare  especially  Galton's  Natural  Inheritance  and  Pearson's 
"  Mathematical  Contributions  to  the  Theory  of  Evolution,"  III,  "  Re- 
gression, Heredity,  and  Panmixia."  (Philosophical  Transactions  of 
the  Royal  Society  of  London,  A,  1896,  Vol.  CLXXXVII.) 

**  The  most  important  works  on  the  correlation  of  individual  char- 
acters have  been  contributed  by  Galton,  Edgeworth,  Pearson,  Weldon, 
and  Yule.  Compare  also  Georg  Duncker,  Die  Methode  der  Variations- 
statistik    (1889),  II,  "Correlation." 

""  See  also  the  references  given  on  p.  367.  Karl  Pearson's  Grammar 
of  Science  contains  a  very  clear  explanation  of  the  correlation  of 
individual  measurements. — Tbanslatob. 


APPENDIX  II 

QUETELET'S  "AVERAGE  MAN" 

Quetelet  declares  in  his  Physics  of  Society  that  he 
has  undertaken  the  task  of  determining  the  man  "  who  is 
to  society  what  the  center  of  gravity  is  to  bodies."  This 
is  the  "  average  man,"  a  fictitious  being  *'  in  whom  all 
processes  correspond  to  the  average  results  obtained  for 
society,"  "  the  mean  about  which  the  elements  of  society 
oscillate."^  Quetelet 's  average  man  possesses  in  an  aver- 
age measure  the  physical  characteristics  and  the  men- 
tal attributes  both  of  the  people  and  the  period  which  he 
represents.  All  of  his  characteristics  and  attributes  are  "  in 
a  proper  equilibrium,  in  a  perfect  harmony,  equally  re- 
moved from  exaggerations  and  defects  of  every  sort,  so  that 
he  must  be  regarded  (for  the  period  in  question)  as  the 
type  of  all  that  is  beautiful  and  good. ' '  ^ 

Quetelet 's  average  man  has,  as  is  well  known,  given  rise 
to  much  critical  discussion.  First,  the  opinion  of  Quetelet 
that  the  average  man  represented  the  type  of  the  beautiful 
and  the  good  was  generally  rejected.     The  average  man 

*  Uber  den  Menschen  und  die  Entwncklung  seiner  Fiihigkeiten,  oder 
Versuch  einer  Physik  der  Gesellsehaft,  German  edition  by  Riecke, 
Stuttgart,  1838,  p.  15. 

^  Ibid.  p.  575.  On  p.  570  Quetelet  expressed  himself  as  follows: 
"  If  the  average  man  could  be  completely  determined,  then,  as  I 
have  already  remarked,  he  could  be  considered  as  a  type  of  the 
beautiful;  and  all  considerable  deviations  from  his  proportions  and 
from  his  qualities  and  capacities  are  to  be  ranked  as  deformities 
and  disease;  whatever  feature  is  not  only  dissimilar  to  the  propor- 
tions and  forms  corresponding  in  him,  but  is  still  more  extreme 
than  the  cases  observed,  would  be  ranked  as  a  monstrosity." 

373 


374  APPENDIX  II 

can  by  no  means  be  regarded  as  physically  beautiful. 
* '  The  average  color  of  the  eyes  would  not  meet  the  demands 
of  beauty,  and  the  average  profile  would  surely  be  far 
removed  from  the  ideal;  besides  in  the  majority  of  men 
the  physical  characters  fall  short  of  the  standard  of 
beauty  (round  shoulders,  flat  chests,  warts,  and  excres- 
cences)." ^"^* 

Just  as  little  can  the  average  man  be  regarded  as  the 
ideal  moral  type.  He  possesses  all  the  good  moral  attri- 
butes only  in  the  average  measure,  but  besides  these  also 
all  the  bad  moral  attributes  in  a  certainly  not  inconsider- 
able measure.  Quetelet  thinks,  to  be  sure,  that  "  an 
attribute  of  man  becomes  a  virtue  when  it  is  a  golden  mean, 
which  is  equally  far  removed  from  all  extravagances  and 
which  is  between  the  limits  marking  the  beginnings  of  vice." 
But  this  view  does  not  take  into  consideration  the  numerous 
reprehensible  human  attributes.  Quetelet 's  praise  of  moral 
mediocrity  would  only  be  justified  if  it  were  established  that 
an  excess  of  the  average  of  one  good  quality  were  insepar- 

*  Westergaard,  Die  Grundziige  der  Theorie  der  Statistik,  p.  276. 
See  also  J.  Bertillon,  Cours  €lementaire  de  Statistique  adminis- 
trative, p.  117,  and  A.  de  Foville,  "Homo  medius  "  (a  paper  read 
before  the  Xlth  Session  of  the  Intern.  Statis.  Institute  in  1907, 
and  printed  in  the  Bull,  de  I'Inst.  int.  de  Stat.,  Vol.  XVII,  p.  46). 
In  opposition  to  these  authors  G.  Viola  has  recently  taken  Quetelet's 
position,  i.  e.,  that  the  average  man  corresponds  in  his  physical  pro- 
portions to  the  ideal  of  beauty.  ( "  La  teoria  dell'  '  uomo  medio  '  e  la 
legge  dell  variazioni  individuali,"  Rivista  italiana  di  Sociologia,  1906.) 

'a  See  Hankins'  very  interesting  study  of  Quetelet  as  Statistician 
(published  as  a  Columbia  University  Study  in  History,  Economics, 
and  Political  Science).  In  this  study  Quetelet's  position  is  set  forth 
and  criticised.  Joseph  Jacobs  has  published  two  papers  in  which 
he  attempts  to  build  up  an  average  Englishman  and  an  average 
American.  ( See  "  The  Middle  American "  in  the  American  Maga- 
zine for  March,  1907,  and  "  The  Mean  Englishman "  in  the  Fort- 
nightly Review,  Vol.  LXXII,  p.  53.)  Mr.  Jacobs  assigns  his  average 
man  a  birthplace  and  early  history,  a  household  budget,  an  occupa- 
tion, definite  personal  qualities,  religion,  political  affiliation,  etc. 
— Tbanslatoe. 


APPENDIX  II  375 

ably  united  with  a  defect  in  some  other  good  quality  or 
with  the  excess  of  a  bad  quality. 

The  question  now  arises,  what  value  may  be  given  the 
average  man  for  special  statistical  purposes  apart  from  his 
alleged  significance  as  a  type  of  the  beautiful  and  the  good  ? 
Quetelet's  average  man  is  in  a  way  the  bearer  of  all  the 
averages  which  may  be  determined  statistically  for  a  defi- 
nite population  and  a  definite  time.  By  means  of  him 
comparisons  of  different  countries  and  times  may  be  estab- 
lished; and,  furthermore,  by  the  comparison  of  individual 
cases  with  the  average  man  a  standard  for  the  judgment 
of  these  cases  may  be  obtained.  Thus,  Quetelet  thinks  that 
in  medicine  "  the  consideration  of  the  average  man  is  im- 
portant to  this  extent  that  it  is  almost  impossible  to  judge 
the  condition  of  an  individual  without  comparing  it  with 
that  of  a  fictitious  being  who  is  regarded  as  normal  and 
who  is  in  reality  nothing  but  the  average  man  whom  we 
have  in  mind.  A  physician  is  called  in  to  see  a  patient 
and  finds  upon  examination  that  the  pulse  is  too  quick 
or  the  respiration  too  hurried,  etc.  It  is  evident  that  when 
such  a  judgment  is  made,  we  recognize  that  the  observed 
phenomena  diverge  not  only  from  those  of  the  average 
man  or  man  in  the  normal  state,  but  that  they  pass  the 
danger  limits.  Every  physician  in  making  such  a  judg- 
ment relies  upon  the  data  in  the  possession  of  science  or 
upon  his  own  experience,  that  is  to  say  upon  a  computation 
of  the  kind  which  we  wish  to  see  carried  out  on  a  larger 
scale  and  with  greater  exactness." 

Quetelet  has  correctly  apprehended  the  principal  purposes 
of  averages :  to  make  comparisons  possible  and  to  afford  a 
standard  for  the  judgment  of  individual  cases.  But  in  all 
statistical  comparisons  we  have  to  do  with  some  single 
observation  element,  and  the  comparison  is  accomplished 
by  relating  two  individual  averages  or  relative  numbers — 
for  example,  the  average  stature  of  the  inhabitants  of  two 
countries,  or  the  rate  of  mortality  in  two  countries.     Simi- 


376  APPENDIX  II 

larly,  in  the  judgment  of  an  individual  case  we  have  to  do 
with  a  single  definite  observation  element — for  instance,  the 
determination  as  to  whether  the  stature  or  the  length  of 
life  of  a  particular  individual  is  above  or  below  the  average, 
and  if  so,  how  much.  Hence,  while  the  comparison  of  in- 
dividual averages  and  their  application  as  a  standard  for 
the  judgment  of  items  have  the  greatest  methodological  sig- 
nificance, on  the  other  hand  the  summing  up  of  all  con- 
ceivable averages  into  one  fictitious  average  man  has  no  real 
statistical  value  whatsoever.  If  we  had  to  compare  the 
average  men  of  two  countries,  we  should  have  to  resolve 
them  into  the  individual  averages  from  which  they  were 
constructed,  and  then  compare  these  averages  with  each 
other.  In  the  same  way,  in  order  to  compare  a  single 
individual  with  the  average  man,  we  should  have  to  con- 
sider one  by  one  the  various  individual  characters  (stature, 
length  of  life,  etc.)  and  to  estimate  the  value  of  each  by  com- 
paring it  with  the  average  value  ascribed  to  the  average  man. 

Let  us  inquire  further  whether  the  construction  of 
Quetelet's  average  man,  apart  from  any  practical  utility, 
is  possible  at  all.  We  must  distinguish  here  between  in- 
dividual characters  such  as  stature,  length  of  life,  etc., 
and  statistical  magnitudes  which,  by  their  nature,  express 
not  the  qualities  of  individual  men  but  the  frequency  of 
definite  events  (births,  deaths,  crimes,  etc.)  for  definite 
groups  of  men. 

It  is  not  inconceivable  that  a  man  may  possess  a  number  of 
individual  characters  in  an  average  measure — for  instance, 
average  height,  weight,  muscular  strength,  income,  length  of 
life,  etc.  But  how  would  it  be  in  regard  to  those  characters 
which  not  all  individuals  possess?  For  such  characters 
there  are  no  general  averages  which  refer  to  the  total 
population  and  hence  they  cannot  be  ascribed  to  the  average 
man.  For  example,  an  average  wage  cannot  be  ascribed 
to  the  average  man,  since  many  men  receive  no  wages  at  all. 
The  same  reasoning  would  hold  true  of  the  average  age 


APPENDIX  11  377 

at  marriage.  Accordingly,  the  average  man  cannot  be 
characterized  in  regard  to  many  points,  and  in  these  re- 
spects he  cannot  be  used  either  for  comparisons  or  as  a 
standard  for  the  judgment  of  individual  cases. 

Thus,  the  construction  of  an  average  man  with  individual 
characters  meets  with  great  difficulties,  especially  since 
many  attributes,  such  as  mental  capacities,  cannot  be  meas- 
ured at  all,  though  the  question  is,  in  this  case,  debatable. 
It  is,  however,  quite  impossible  to  equip  this  average  man 
with  those  averages  which  express  the  mean  frequency  (or 
probability)  of  certain  events  (frequency  of  births,  crimes, 
suicides,  probability  of  death,  etc.) .  These  are  values  which 
are  produced  by  interrelating  statistical  masses  and  have 
a  meaning  only  in  connection  with  the  masses  from  which 
the  events  in  question  proceed.  If,  for  instance,  the  fre- 
quency of  crime  amounts  on  the  average  to  12° /^^  for  the 
entire  population,  this  means  that  of  10,000  inhabitants 
12  commit  a  crime  in  a  year.  But  the  "  average  man  " 
either  commits  a  crime  or  does  not.  In  the  first  case  his 
average  would  be  1,000  Voo?  i^  the  second  OVoo-* 

Mention  should  also  be  made  of  the  question,  broached 
by  some  authors,  as  to  whether  a  man  possessing  all  the 
observation  elements  in  an  average  measure  may  really  exist. 
Quetelet  has  himself  expressed  the  opinion  that  the  perfect 
type  of  the  average  man  is  hardly  within  the  bounds  of 
possibility;  that  in  general  it  is  only  possible  to  attain  the 
type  in  single,  more  or  less  numerous,  respects.^    It  is,  in 

*  The  "  abstract  man "  of  Lexis  is  to  be  distinguished  from  the 
"  average  man."  ( See  "  Ubersieht  der  demographischen  Elemente," 
Bull,  de  I'Inst.  de  Stat.,  Vol.  VI,  p.  40,  and  Abhandlungen,  p.  60.) 
The  former  is  not  characterized  by  definite  attribut€S,  but  he 
merely  functions  as  the  bearer  of  various  demographic  probabilities. 
It  is,  according  to  Lexis,  the  final  goal  of  demography  to  compass 
the  life  history  of  man,  considered  in  the  abstract,  v.  Bortkiewicz 
designates  Lexis'  "  abstract  man "  to  be  a  revised  and  improved 
edition  of  the  "average  man."  (Conrad's  Jahrb.,  3rd  series,  Vol. 
XXVII,  p.  245.) 

"  liber  den  Menschen,  etc.,  p.  576. 


378  APPENDIX  II 

truth,  highly  improbable  that  an  individual  should  corre- 
spond to  the  average  measure  in  various  respects  at  once. 
"Whether  the  averages  of  two  observation  elements  will 
coincide  more  or  less  often  in  the  same  individuals  depends 
essentially  upon  the  extent  of  the  correlation  between  those 
elements. 

As  previously  mentioned,®  two  individual  characters 
are  in  correlation  if  their  sizes  are  related.  In  case  of 
perfect  correlation  between  two  characters  the  averages 
must  appear  simultaneously;  if  two  characters  are  meas- 
ured in  the  same  individuals  (for  instance,  the  length 
of  two  different  parts  of  the  body),  then,  in  case  of  perfect 
correlation,  those  individuals  who  have  one  character 
of  average  size  must  also  have  the  other  of  average  size. 
If  there  is  only  a  partial  correlation  between  the  two  char- 
acters the  averages  need  not  appear  simultaneously  in 
all  cases,  but  there  is  an  increased  probability  of  such  ap- 
pearance and  the  degree  of  this  probability  depends  upon 
the  degree  of  correlation. 

Now,  in  fact,  in  biology  and  anthropology  there  is  gen- 
erally at  least  a  partial  correlation  between  various  char- 
acters and,  therefore,  there  is  an  increased  probability 
for  the  simultaneous  appearance  of  the  averages  of  these 
characteristics.  On  the  other  hand,  there  is  a  lack  of 
correlation  between  demographic  and  economic  elements 
of  observation  as  well  as  between  these  and  anthropological 
elements.^  Of  course  no  one  will  claim  that  length  of  life 
increases  in  a  definite  ratio  with  bodily  size  or  with  age  at 
marriage.  There  is,  therefore,  no  increased  probability  for 
the  simultaneous  appearance  of  the  various  averages  of 

•  Compare  p.  370  f. 

*  The  only  known  illustration  of  an  undoubted  correlation  offered 
by  demography  is  the  combined  ages  of  those  marrying.  But  in  this 
case  we  are  not  concerned  with  two  characters  of  the  same  individual, 
but  by  the  paired  observations  of  different  individuals,  namely,  one 
individual  of  each  sex  contracting  marriage. 


I 


APPENDIX  II  379 

these  elements.  People  of  average  height  will  not  be  found 
relatively  more  often  among  those  of  average  length  of 
life  than  among  those  of  any  other  age.  It  follows  from 
this  that  people  who  unite  even  a  few  averages  for  different 
observation  elements  (not  exclusively  in  anthropology)  are 
quite  exceptional. 

This  fact  must  unquestionably  diminish  the  value  of  the 
* '  average  man. ' '  "While  single  averages  may  often  possess  a 
typical  character,  that  is,  may  be  looked  upon  as  normal 
values,  from  which  the  concrete  single  cases  only  diverge 
because  of  accidentally  disturbing  causes,  the  average  man 
who  combines  all  the  averages  can  by  no  means  be  looked 
upon  as  a  typical  normal  man  from  whom  all  other  men 
only  diverge  accidentally.  He  is  an  abstraction  without 
any  foundation  in  fact  and  has  no  independent  methodo- 
logical' value. 

The  "  average  man  "  has  evidently  sprung  into  being 
rather  too  hastily  from  Quetelet's  generalizations  upon  the 
results  obtained  by  him  in  anthropology.  If  all  observa- 
tion elements,  like  measurements  of  height,  showed  a  sym- 
metrical distribution  about  a  typical  mean  and  if,  besides, 
there  were  a  considerable  degree  of  correlation  between 
the  various  observation  elements,  then  the  average  man 
would  be  indeed  a  good  representative  of  the  prevail- 
ing characters.  But  since  these  two  presuppositions  are 
not  fulfilled,  the  theory  of  the  average  man  is  hardly  more 
than  a  historical  reminiscence,  of  interest  only  in  connec- 
tion with  the  personality  of  its  author.  In  modern  statis- 
tics, which  emphasizes  specialization  and  detail  work,  the 
average  man  cannot  have  any  additional  significance.  Aver- 
age values  for  whole  nations  are  seldom  useful  for  scientific 
purposes;  the  great  object  is  to  obtain  values  for  smaller 
masses,  definitely  characterized  and  as  homogeneous  as 
possible,  in  order  to  form  a  basis  for  comparisons  or  for 
other  scientific  investigations. 


APPENDIX  III 

REFERENCES 

The  following  are  the  more  important  systematic  and 
general  methodological  works  on  statistics,  inclusive  of  gen- 
eral works  on  population  statistics: 

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Bennini,  R.,  Principii  di  Statistiea  methodologiea,  Turin,  1906. 

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Bertillon_,  Jaques,  Cours  elementaire  de  Statistique  adminis- 
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Block,  M.,  Traite  theorique  et  pratique  de  Statistique,  2nd  Ed., 
1886. 

Block-Scheel,  Handbueh  der  Statistik,  Leipsic,  1879. 

BowLEY,  Arthur  L.,  Elements  of  Statistics,  3rd  Ed.,  1908. 

BowLEY,  Arthur  L.,  An  Elementary  Manual  of  Statistics,  Lon- 
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Cheysson,  E.,  Les  methodes  de  la  Statistique.  (Reprint  from  the 
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C0LAJANNI_,  Napoleone,  Statistica  e  demografia: 
I.     Statistiea  teorica,  1904. 
II.     Demografia,  1904. 

Conrad^  J.,  Grundriss  zum  Studium  der  politischen  Okonomie, 
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der  Statistik;  Die  Bevolkerungsstatistik.  Part  II:  Statistik 
der  wirtschaftlichen  Kultur;  1st  Division:  Berufsstatistik, 
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Gewerbstatistik. 

Davenport,  C.  B.,  Statistical  Methods,  New  York,  1904. 

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Elderton,  W.  Palin,  Frequency- Curves  and  Correlation,  London, 
1906. 

Elderton,  W.  p.  and  E.  M.,  Primer  of  Statistics,  1909. 

Fallati,  Einleitung  in  die  Wissenschaft  der  Statistik,  Tiibingen, 
1843. 

380 


APPENDIX  III  381 

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Turin,  1891. 
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sic,  1898. 
FlecheY;,  E.,  Notions  generales  de  Statistique,  Paris,  1872. 
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Political  Economy. 
Franchi,  L.,  Appunti  di  statistica,  Modena,  1898. 
Gabaglio,  Antonio,  Storia  e  teoria  generale  della  statistica,  Milan, 

1880. 
Garnier,  Joseph,  Elements  de  Statistique,  contained  in  Notes  et 

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Guillard,  Achille,  Elements  de  Statistique  humaine  ou  demo- 

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Jacquart,  Camille,  Statistique  et  science  soeiale,  Brussels,  1907. 
Jahnson,  Theory  of  Statistics,  2nd  Ed.,   St.  Petersburg,  1887 

(Russian). 
JULIN,  Armand,  Precis  du  Cours  de  Statistique,  Paris,  1910. 
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Vienna,  1904. 
Haushofer,  Max,  "Bevolkerungslehre"  in  Natur  und  Geistes- 

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Lexis,  W.,  Article  on  "  Statistik  (1  Allgemeines)  "  in  the  Hand- 

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Lexis,  "W.,  Zur  Theorie  der  Massenerscheinungen  in  der  menseh- 

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Liesse,  Andre,  La  Statistique,  ses  difficultes,  ses  proeedes,  ses 

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382  APPENDIX  III 

Matheson,  R.  E.,  The  Mechanism  of  Statistics,  Dublin,  1889. 

Mayo-Smith_,  Richmond_,  Article  on  "  Statistical  Method "  in 
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by  The  Maemillan  Co.,  for  the  Am.  Econ.  Ass'n. 

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INDEX 


Abstract  man,  Lexis  on,  377; 
see  Average  man 

Accidental  errors,  166;  see  Error 

Adjustment,  references,  147; 
mortality  tables,  348 

Ages,  characteristic  conforma- 
tion of,  346 

Ammon,  281,  287 

Analytic  probability,  20 

Anthropology,  typical  series, 
276;  stability  of  statistical 
data,  298 

Anthropometry,  see  Anthropol- 
ogy 

Arithmetic  mean,  8;  estimation 
of,  35;  length  of  life,  40  f.; 
length  of  marriage,  44  f . ;  con- 
cept and  qualities  of,  138;  of 
the  three  groups  of  series, 
139  f. ;  computation  of,  142 ; 
adjustment  or  graduation  of 
series  before  eompviting,  146  f. ; 
application  of,  148;  graphic 
representation,  150;  in  math- 
ematical statistics,  163  f.;  in 
theory  of  probability,  171  f.; 
accuracy  of,  171;  scientific 
value  of,  175;  as  normal  value, 
178;  typical,  178  f.;  signifi- 
cance of  a  diff"erence,  186 

Array,  items  arranged  in  order 
of  magnitude,  203 

Asymmetry,  282  f. ;  see  Disper- 
sion, Characteristic  conforma- 
tion, Gaussian  law 

Average  man,  373  f. ;  and  cor- 
relation, 378 

Averages,  works  on,  2  f . ;  impor- 
tance of,  3 ;  most  probable  size, 
11;  typical,  12;  subjective  and 
objective,  10,  13;  for  series  of 
first  group,  7  f . ;  for  series  of 
second  group,  15  f.;  for  series 
of  third  group,  18  f.;  isolated. 


25  f . ;  nature  and  purpose  of, 
92  f. ;  comparison  of,  93  f. ; 
homogeneity  in  comparison  of, 
101  f.;  investigation  of  causes, 
110;  as  a  standard  for  judg- 
ing items,  122  f.;  as  a  basis 
for  measuring  dispersion, 
125  f.;  the  varieties  of,  131  f.; 
of  logarithms,  135;  terminol- 
ogy, 135  f.;  the  arithmetic 
mean,  138  f.;  the  geometric 
mean,  194  f.;  the  median, 
199  f.;  the  mode,  222  f. 

Babbage,  347 

Babson,  101 

Bailey  and  Parmelee,  84 

Ballod,  358 

Bernoulli,  171;  theorem  of,  175 

Bertillon,  A.,  2,   10,   19,  80,   LS4, 

228  f . ;      system     of     detecting 

criminals,  228 
Bertillon.   J.,   88,    120,    181,   184, 

230.  280,  361,  369,  374 
Beveridge,   364 
Bibliography,  380  f. 
Binomial  expansion,  272.  288 
Birth   rates,    107  f.;    stabilitv  of, 

299 
Blaschke,  51,  147,   164,  277,  323, 

329 
Block,   37,   120,   181.   184 
Bodio,  305.  312 
Boeckh.  45.  52.  54,  201 
Boehmert.  26.  131,  133 
Bohlmann.  329 
Booth,   159 
Bortkiewicz,    von,    20,    132,    146, 

160,    164.    174,    176,    177,    180, 

190.    323.    324,   327.    331,    334, 

336  f..  348.  350  f.,  377 
Bowley,  1.  16.  92.  97.  98.  136, 

147,'  158,  162.  164.  171.  190, 

199,  206,  212  f.,  220,  221,  233, 


385 


386 


INDEX 


234,     235  f.,     242,     243,     260, 

272,    291,    338,    348,    363,    366, 

368 
Bravais,  367 
Brookmire,   101 
Bruns,  287 
Buckle,  305 
Budgets,   family  and  individual, 

27 
Buzek,  366 

Causation,  investigation  by  com- 
parison of  averages  and  rela- 
tive numbers,  110  f.;  caution 
as  to  conclusions  concerning, 
120;  comparison  of  averages 
before  and  after  an  event,  121; 
investigation  by  means  of  se- 
ries of  characteristic  con- 
formation, 352;  series  of  third 
group,  353;  illustrations  as  to 
conclusions  concerning,  358; 
geographic  and  time  series, 
359  f. ;  geographic  series,  360 ; 
time  series,  362;  see  Correla- 
tion 

Census,  U.  S.  method  of  comput- 
ing isolated  average,  26;  dura- 
tion of  marriages  from,  47; 
marital  fecundity  from,  53; 
see  Wages,  Ages,  Population 

Cephalic  index,  definition  of,  19; 
evolution  of,  281 

Chance,  see  Probability,  Error, 
Binomial  expansion 

Characteristic  conformations, 

time  series,  294;  series  having, 
341  f.;  evolutionary  series, 
342;  for  series  of  individual 
observations,  345  f . ;  quantita- 
tive series,  346;  and  causation, 
352;  see  Periodic  series.  Evo- 
lutionary series.  Crises 

Charlier,  287 

Classes,  formation  of  magnitude, 
80  f. ;  shaded  maps,  87  f. ; 
may  lead  to  new  series,  88; 
cumulative,  85,  87,  238 

Classification,  of  series,  7  f.,  24 

Class-interval,  see  Classes 

Coefficients,  statistical,  16 

Colajanni,  32,  118,  136,  204,  224, 
273 


Collective  objects,  10 

Combination  mean,  134 

Communicants,  estimation  of 
number,  40 

Comparison  of  relative  numbers 
and  averages,  homogeneity  a 
necessity  for,  101  f.,  106  f.;  see 
Causation 

Composition,  method  of,  282 

Conrad,  157 

Constancy  of  small  numbers, 
335  f. 

Constituents  of  greater  totality, 
homogeneity,  72  f. 

Contraharmonic  mean,  132 

Coordinate  numbers,  16,  21,  28 

Corradi,   305 

Correlation,  between  time  series, 
363  f.;  coefficient  of,  367  f.; 
formula  for  coefficient  of,  368; 
between  individual  characters, 
370  f . ;  biological  and  anthro- 
pological data,  371;  regression, 
372;  average  man,  378;  see 
Causation 

Correlation  table,  370 

Crises,  periodicity  of,  345;  ef- 
fect on  demographic  statistics, 
295  f.;  statistics,  295  f.;  sta- 
tistical barometers  of,  101 

Crude  rates,  argument  for,  75; 
defects  of,  107  f.;  see  Homo- 
geneity, Refined  rates 

Cumulative  classes,  85 ;  curve 
from,  87 ;  computation  of  mode 
from,  238 

Curves,  multimodal,  280;  Pear- 
son's types,  289;  see  Graphic 
representation 

Customary  item,  see  Mode 

Czornig,  26 

Czuber,  20,  24,  147,  164,  174, 
192,  199,  201,  222,  240,  323, 
325,  329  f.,  348,  374 

Davenport,  288 

Davoisier,  37 

Death  rates,  28 ;  as  averages,  31 ; 

defects  of  a  general  rate,  109; 

stability  of,  299;  see  Mortality 

table 
Deciles,     use     in     formation    of 

classes,   89  f . ;    use  in  measur- 


INDEX 


387 


ing  dispersion,  258;  see 
Median 

Demographic  statistics,  factors 
influencing,  31  f.;  comparison 
of  relative  numbers,  107;  see 
Birth  rates,  Death  rates.  Popu- 
lation, Census 

Deparcieux,  42 

Determinism,  see  Free  will 

Deviation,  from  arithmetic  mean, 
264  f. ;  standard  deviation, 
265;  see  Dispersion 

Dewey,  89 

Difference,  significance  of  a,  186; 
illustrations,   189 

Dispersion,  of  repeated  measure- 
ments, 13;  use  of  averages  in 
measuring,  125  f.;  about  the 
mean,  249  f.;  purpose  and 
meaning  of  establishing,  251; 
for  typical  series,  252;  of  in- 
dividual observations,  256; 
range  as  a  measure  of,  257; 
indicated  by  difference  between 
averages,  258;  indicated  by 
quartiles,  deciles,  percentiles, 
258;  Bowley's  index  of,  260; 
Statistical  Congress  concern- 
ing, 261;  relative  to  average 
of  items,  261  f.,  284;  deviations 
from  arithmetic  mean,  264; 
above  and  below  mean,  26G; 
of  irregular  series,  267  f.,  295; 
necessity  of  homogeneous  se- 
ries, 268  f . ;  theory  of  error, 
270  f. ;  mathematical  methods 
of  expressing,  274  f. ;  range  as 
a  measure  of,  292;  of  series  of 
second  group,  292  f. ;  of  serips 
of  characteristic  conformation, 
294;  of  series  of  third  group, 
297  f. ;  time  series,  297  f. ;  sta- 
bility of  series,  300  f.;  of  cer- 
tain series  of  relative  numbers 
and  averages,  313  f.;  elemen- 
tary mathematical  methods  of 
measuring,  second  group,  316; 
of  series  of  numerical  proba- 
bilities, 319;  normal,  supra- 
normal  and  infra-normal,  322; 
see  Stability,  Deviation 

Distribution,  normal,  11,  12;  of 
wealth  and  incomes,  248  f . ;  see 


Frequency  tables.  Series,  Dis- 
persion 

Dormoy,  324 

Drobiseh,  306 

Duncker,  263,  280,  288,  289,  290, 
371 

Dupin,  42 

Earnings,  usage  in  the  U.  S., 
26  f . ;  see  Wages 

Edgeworth,  1,  2,  24,  96,  97,  133, 
164,  188,  189,  196,  271  f., 
278  f.,  281,  282,  285,  286.  287, 
289,  312,  347,  351,  367,  372 

Elderton,  W.  P.,  288;  and  Elder- 
ton,  E.  M.,  367 

Engel,  37,  293 

Error,  use  of  arithmetic  mean  in 
theory  of,  164  f.;  probable,  166; 
mean,  166;  Gaussian  function 
of,  166;  dispersion,  270  f.;  ap- 
plication of  theory  to  mortal- 
ity, 276  f.;  application  to 
crops,  277;  generalized  law  of, 
278  f.;  skew  curves  of,  286  f.; 
Edgeworth's  generalized  law 
of,  288;  method  of  computing 
probable,  323;  normal-acci- 
dental and  physical  compo- 
nents of,  332;  coefficient  of 
correlation,  368 

Errors,  systematic  and  acci- 
dental, 170  f. 

Essars,  des,  349 

Estimated  averages,  33  f. 

Euler,  343 

Evolutionary   series,   342 

Expectation  of  life,  41  f.,  183 

Falkner,  99 

Farr,  53 

Fechner,  2,  10,  80,  82,  131,  133, 
134  f.,  146,  164,  169,  199,  202, 
205,  210  f..  212,  238,  259,  262, 
265,  272,  283  f. 

Fecundity,  marital,  50  f.;  table 
of,  56  f.;   sterile  marriages,  67 

Fertility,  see  Fecundity 

Fisher, 'll4 

Fluctuation,  165  f.;  see  Disper- 
sion 

Flux,  133,  198 

Formulas,   empirical,   279;    posi- 


388 


INDEX 


tions  of  Pearson  and  Edge- 
worth  concerning,  291 

Foville,  49,  349 

Fox,  217 

Free  will,  and  statistical  regu- 
larity, 305 

Frequency  tables,  individual  ob- 
servations, 9 ;  cumulative 
classes,  85;  computation  of 
arithmetic  mean  from,  142  f.; 
median,  206;  mode,  235  f.;  see 
Magnitude  classes 

Galton,  87,  89,  133,  164,  210,  212, 
220,  238,  287,  351,  367,  371, 
372 

Gaussian  law,  repeated  measure- 
ments, 12;  formula,  166;  ap- 
plication to  statistics,  167  f.; 
measures  of  dispersion,  271; 
asymmetrical,  282  f.;  anthro- 
pometry, 276;  logarithmic  gen- 
eralization, 283;  as  special 
case  of  general  law,  284;  see 
Probability,  Error,  Distribu- 
tion, Dispersion 

Geissler,  89,  115,  325 

Generalized  law  of  error,  278  f. 

Generation,  definition  and  statis- 
tical determination  of  length, 
48  f. 

Genetic  probability,  20 

Geometric  mean,  133,   194 

Giffin,  99,  161,  163 

Gompertz,  347 

Graduation  of  tables,  see  Adjust- 
ment. 

Graphic  representation,  shaded 
maps,  87  f.;  arithmetic  mean, 
149  f. ;  use  in  finding  median, 
212  f. ;  use  in  finding  mode, 
237  f. ;  multimodal  curves,  de- 
composition of,  280;  skew 
curves,  286  f.;  historical 
curves,  363 

Gryzanovski,  112 

Guerry,  1,  164 

Guillard,   361 

Gulick,  89 

Gumplowicz,  36 

Hankins,  306,   374 
Harmonic  mean,  132 


Haushofer,  46,  52,  181,  185 

Herri,  326 

Holmes,  2 

Homogeneity     of     series,     65  f.; 

problem   of    null    items,    66  f. ; 

problem    of    categories,    70  f. ; 

erroneous      conclusions      from 

comparing        non-homogeneous 

masses,  105 
Hooker,  363,  367 

Inama-Sternegg,  von,  49,  78,  118 

Income,  estimation  of  national, 
35 ;  Pareto's  formula,  348  f . ; 
see  Wages 

Index  numbers,  mean,  95  f.;  of 
prices,  96;  references,  97,  99; 
of  wages,  98  f. ;  of  consump- 
tion, 99 ;  of  economic  and  so- 
cial conditions,  100  f. ;  weights, 
156  f.;   geometric  mean,  196 

Individual  observations,  7f. ;  ho- 
mogeneity, 66  f . ;  null  items 
among,  102;  see  Series 

Induction,  distinguished  from 
statistical  method,  112 

Industrial  condition,  effects  on 
social  and  economic  statistics, 
304;    index  numbers  of,   100  f. 

Interdependence  of  events,  114, 
115;  see  Causation,  Correla- 
tion 

Isolated  averages,  25  f. ;  as  mere 
ratios,   28 

Items,  logical  agreement  of, 
60  f.;  see  Individual  observa- 
tions. Series,  Frequency  tables 

Jacobs,  374 

Jenkins,  365 

Jevons,  118,  133,  195  f.,  343 

Juglar,  343,  365 

Kaans,  329 

Kammann,   327 

Kapteyn,  287 

Karup,  147 

Keynes,  112 

Kiaer,  54,  58,  68 

Knapp,  164,  306 

Kollman,  318 

Korosi,  50,  55,  58,  160,  351,  358 

Kries,  von,  164 


INDEX 


389 


Lambert,  347 

Laplace,  365 

Laughlin,  114 

Law,  of  great  numbers,  172,  174; 
exponential  law  of  great  num- 
bers, 288;  nature  of  statis- 
tical, 302;  of  small  numbers, 
335  f.;  see  Gaussian  law.  Dis- 
persion, Free  will,  Causation, 
Correlation 

Lawrence,  355 

Lazarus,  347 

Lehr,  331 

Leibnitz,  63 

Length  of  life,  see  Expectation 
of  life,  Mortality  table 

Lexis,  19.  21,  51,  117,  164,  176, 
181,  185,  191,  201,  202,  216, 
217,  220,  221,  222,  226,  229, 
231,  240,  241,  261,  270,  274  f., 
276,  281,  291,  300,  305,  306, 
316,  322  f.,  324,  326,  328,  342, 
350  f.,  374,  377 

Liesse,  200 

Litrow,  347 

Logarithmic  mean,  135;  Gauss- 
ian law,  283  f. ;  see  Geometric 
mean 

Magnitude  classes,  formation  of, 
80  f . ;  see  Classes,  Frequency 
tables 

Magnitudes,  see  Measurements, 
Items 

Makeham,  mortality  formula, 
347 

Malthus,  42,  52,  343 

Mandello,  218,  243 

March,  58,  238,  259,  349,  366 

Marriage,  rates,  107  f.;  influence 
on  mortality,  120  f.;  average 
length  of,  44  f. 

Marshall,  347 

Mass,  use  of  term  in  statistics, 
14 

Mataja,  39 

Mathematical  statistics,  refer- 
ences, 164;  importance  of, 
179  f.;  value  of  formulas, 
347  f.;  see  Gaussian  law, 
Probabilitv,  Error,  Correlation 

Mayo-Smith,'  317  f. 

Mayr,  von,  10,  40,  48,  56,  75,  77, 


80,  87,  92,  104,  117,  143,  144, 
150,  161,  183  f.  261,  263  f., 
293,  310,  312,  316,  343,  346, 
360,  362,  365 

McAlister,  133 

Mean,  see  Average,  Arithmetic 
mean.  Median,  Mode,  Contra- 
harmonic  mean.  Harmonic 
mean.  Geometric  mean.  Com- 
bination mean 

Mean  index  numbers,  see  Index 
numbers 

Measurements,  repeated  of  same 
object,  11;  continuous  and  dis- 
continuous, 81;  see  Items, 
Series 

Median,  defined,  8  f . ;  concept 
and  properties  of,  199  f. ;  prob- 
able value,  201 ;  where  possi- 
ble, 202;  determination  of, 
205  f. ;  hypothesis  of  uniform 
distribution  in  central  class, 
207;  Galton's  use  of.  210; 
graphic  method  of  obtaining, 
212  f.;      application      of      the, 

215  f.;     lifetimes.     215;     ages, 

216  f.;  wages,  217  f.;  anthro- 
pometric data,  219;  prices, 
219;  errors,  219  f. 

Meitzen,  185,  305 

Messedaglia,  2,  125,  132,  196  f., 
305 

Meteorology,  stability  of  data, 
298  f. 

Method  of  difference.  111 

Mill,  J.  S.,  Ill 

Mischler,  30,  47,  48,  77 

Mitchell,  86,  99 

Mitscherlich,  277 

Mode,  defined,  9;  estimation  of, 
34;  advantages  and  limita- 
tions of,  38;  concept  and  prop- 
erties of,  222  f. ;  as  typical 
mean,  226;  secondary  values, 
230;  determination  of,  233  f.; 
graphic  method  of  finding, 
237  f.;  applications  of,  240  f.; 
lifetimes.  240;  wages,  242  f.; 
prices,  244;  other  terms  for, 
245;  ease  of  estimating.  246  f. ; 
decomposition  of  multimodal 
curves,  280 

Modulus,  165  f. 


390 


INDEX 


Moore,  14,  86,  112,  114,  118,  269 

Morpurgo,  305 

Mortality  and  marriage,  120  f.; 
standard,  160  f.;  significance  of 
difference  in,  191,  192;  by  age 
groups,  229;  stability  of, 
328  f. ;  see  Mortality  tables 

Mortality  tables,  definition  of, 
43;  modes  in,  231;  Austrian, 
241 ;  symmetric  distribution 
in,  276  f. ;  decomposition  of 
data  of,  281;  mathematical 
formulas  for,  347  f . ;  see  Ad- 
justment, Mortality 

Moser,  56,  347 

Mulhall,  97 

Natality,  bigenous  table  of,  50 

Natural  districts  and  time  pe- 
riods,  77 

Nearing,   151 

Neefe,  326 

Neumann-Spallart,  100 

Newsholme,  147 

Normal  curve  of  errors,  see 
Gaussian  law.  Probability,  Er- 
ror, Binomial  expansion 

Norton,  344 

Null  items,  problem  of,  66  f . ; 
elimination  of,  74  f.,  102;  see 
Homogeneity  of  series 

Oettinger,  31,  232,  292,  310 
Ogle,  160 
Opperman,  347 

Order  of  items,  see  Array, 
Median 

Pal  grave,  97 

Pareto,  348  f. 

Pearson,    164,    274,    277,    280  f., 

288  f.,  319,  367,  368,  372 
Peek,  328 
Percentiles,  use  in  formation  of 

classes,  89  f.;   defined,  199;   as 

measure  of  dispersion,  258 
Periodic  series,  344  f. 
Permanence    of    small    numbers, 

335  f. 
Persons,  349,  367 
Pierson,  96 
Poisson,  171 
Pokrovsky,  366 


Population,  estimation  of,  35  f., 
40;  standard,  159  f.;  com- 
pound interest  law,  343;  by 
ages,  346 

Porter,  272 

Power  means,  134 

Precision,  measure  of,  13,  165  f.; 
see  Dispersion 

Predominant  item,  see  Mode 

Prevailing  item,  see  Mode 

Price,  42 

Prices,  index  numbers  of,  96  f. 
weighted     indices     of,     156  f. 
geometric  mean  index  of,  196 
mode  of,  244;  see  Index  num- 
bers 

Prinzing,  68 

Probability,  statistical,  19;  ana- 
lytic or  secondary,  20;  genetic 
or  primary,  20;  functions,  20; 
empirical,  172  f.;  objective, 
320  f;  empirical  values  of, 
320  f.;  theory  of,  163  f.,  166, 
177  f.,  276  f.,  319,  320,  335  f.; 
Pearson's  curves,  288  f. 

Probable  error,  166,  201 

Probable  item,  see  Median 

Production,  estimation  of,  36 

Quartiles,  defined,  199;  as  meas- 
ure of  dispersion,  258  f. 

Quetelet,  3,  31,  49,  132,  169,  184, 
230,  253,  254,  276,  294,  299, 
305,  338,  346,  347,  373  f. 

Quetelet's  average  man,  373  f. 

Rainfall,  method  of  computing 
average,  159 

Range  of  a  series,  77  f.;  as  a 
measure  of  dispersion,  257 

RSth,  55 

Refined  rates,  108  f. 

Regression,  372;   see  Correlation 

Relative  numbers,  as  averages, 
30  f. ;  estimation  of,  39 ;  ho- 
mogeneity, 73  f. ;  treatment  of 
null  items,  74  f. ;  comparison 
of,  106  f.;  causation,  110  f.j 
law  of  error,  167  f. 

Rhenisch,  305 

Rosenfeld,  231 

Rubin,  36,  160,  246 

Rumelin,  49,  185,  227,  305 


INDEX 


391 


Sauerbeck,  157,  161 

Schmoller,  306 

Schumpeter,  97 

Scratchlev,  348 

Seailles,  349 

Separation,  method  of,  279,  280  f. 

Series,  classification  of,  7  f .,  24; 
first  group  defined,  7;  second 
group  defined,  14;  third  group 
defined,  16;  homogeneity,  65; 
subdivision  into  more  homo- 
geneous parts,  75  f.;  typical 
and  non-typical,  181,  274  f.; 
asymmetrical,  278  f.;  decom- 
position of,  279  f.;  explanation 
of,  asymmetrical,  285;  unilat- 
eral, 287,  291;  of  character- 
istic conformation,  341  f.;  evo- 
lutionary, 342 ;  periodic,  344  f. 

Sex-ratios,  29;  as  averages,  30; 
stability  of,  298  f.;  reasons 
for  fluctuations,  312,  316;  as 
numerical  probabilities,  324; 
illegitimate  and  still-births, 
325;  multiple  births,  326 

Sheppard,  367 

Significance  of  a  difference,  113, 
186  f.,  189;  application  of  the- 
ory of  probability,  193 

Skew  curves,  286  f. 

Sonnenfels,  40 

Stability  of  series,  298  f.;  nat- 
ural and  social  causes  of, 
300  f. ;  suggested  reasons  for, 
303,  307;  of  suicides,  300,  327, 
335  f.;  effect  of  homogeneity, 
310;  time  series,  297  f.;  geo- 
graphic series,  311  f.;  sex- 
ratios,  324  f.;  death  rates, 
328  f. ;  of  small  numbers, 
335  f . ;  see  Dispersion 

Staehr,  118 

Standard  deviation,  265;  see 
Dispersion 

Standard  mortality,  160  f. 

Standard  population,  159  f. 

Stark,  325 

Statistical  probability,  definition 
of,  19 

Statistics,  definition  of,  1 ; 
method,  112,  116;  mathemat- 
ical and  non-mathematical, 
178  f.;    law,   302 


Subordinate    numbers,    definition 

of,  16,  29 
Suicides,  regularity,  327 
Sundbaerg,   160 
Sussmilch,  52,  299 
Systematic  errors,  166 

Tammeo,  3 

Taussig,  161 

Thiele,  287,  347 

Trade,  estimation  of,  36 

Translation,  method  of,  279, 
285  f. ;  generalized  method, 
287  f. 

Translator's  notes,  14,  16,  26,  83, 
86,  97,  99,  101,  107,  108,  109, 
112,  114,  118,  121,  133,  134, 
147,  151  f.,  162,  163,  166, 
195  f.,  198,  223,  230,  243,  267, 
269,  289,  291,  299,  306,  344, 
346,  349,  355,  363,  364,  367, 
368,  370,  374 

Tsehuprow,  112,  188,  191,  307, 
328 

Turquan,  49 

Types  of  curves,  Pearson's,  289 

Typical  mean,  168,  178  f.,  182; 
criterion  for,  274 

Unit  of  observation,  60;  see 
Homogeneity  of  series 

Vacher,  49 

Venn,  3.   112,  136,  223 

Viola,  374 

Wages,  usage  in  U.  S.,  26  f.; 
estimation  of  mode,  34;  cumu- 
lative classes,  85;  movement 
in  the  U.  S.,  1890-1900,  86; 
index  numbers  of,  98  f.,  158; 
effect  of  non-homogeneity  in 
comparison  of.  103  f.;  arith- 
metic average  of.  151;  median 
of,  208;  mode  of.  235  f.,  242  f. 
Wagner,  32.  293.  295.  306,  317 
Wappaus,    42.   46,   53,    188,   305, 

361 
Weiglited  average,  see  Weights 
Weights,    items    of    third    group, 
17;     use    of,    141;     arithmetic 
mean,    150;    wages,    151;    esti- 


392 


INDEX 


mation  of,  154;  prices,  156  f.; 

importance  of,  161  f. 
Weldon,  288,  372 
Westergaard,    36,    63,    107,    109, 

147,    161,    164,    173,    175,    190, 

191,    192,    197,   323,    327,    347, 

348,  374 
Wittstein,  147,  347 
Wood,  98  f.,  121,  158,  161,  243 


Woolhouse,  147 
Wundt,  302 

Young,  84,  147 

Yule,  108,  164,  367,  371 

Zimmerman,  329 
Zuckerkandl,  97 


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Simply  and  lucidly  tells  what  a  railroad  company  is,  what  it 
does,  and  how  it  does  it. 

MORE'S    WAGE-EARNERS'   BUDGETS 

A  Study  of  Standards  and  Cost  of  Living  in  New  York  City. 
By  Louise  B.  More.  With  a  preface  by  Professor  F.  H.  Gm- 
DINGS.     280  pp.     Svo.     $2.50  net. 

A  report  of  the  first  investigation  carried  on  under  the  direc- 
tion of  the  Committee  on  Social  Investigations  at  Greenwich 
House,  a  social  settlement  on  the  lower  West  Side  of  New  York 
City,  among  workingmen's  families  of  different  races  and  occu- 
pations. 

ZARTMAN:   THE   INVESTMENTS   OF  LIFE  INSURANCE 
COMPANIES 

By  Lester  W.  Zartman,  Instructor  in  Yale  University. 
259  pp.     12mo.     $1.25  net. 

It  analyzes  investments  and  the  earning  power  of  the  various 
assets  of  life  insurance  companies.  The  interest  rate  is  calculated 
by  a  new  and  exact  method.  The  author  also  discusses  the  rela- 
tions of  the  investments  to  social  welfare,  and  the  proper  control 
of  the  immense  assets  of  the  companies. 

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TEXT-BOOKS  IN  ECONOMICS 

ADAMS'S  SCIENCE  OF  FINANCE 

37  Henry  C.  Adams,  Professor  in  the  University  of  Michigan. 
573  pp.     870.     (^American  Science  Series.)     $3.00. 

Edwin  R,  A.  Seligman,  Columbia  University,  in  "Political  Science  Quar- 
terly " :— W/U  at  once  command  attention  as  a  lasting  contribution  to  eco- 
nomic literature.  ...  It  is  perhaps  no  exaggeration  to  say  that  Professor 
Adams  is  at  the  head  of  those  American  scholars  who  have  grasped  the  essen- 
Ual  spirit  of  modern  industrial  life  ;  and  it  is  likewise  no  exaggeration  to  claim 
for  this  volume  the  distinction  of  being  one  of  the  most  original,  the  most  sug- 
gestive, and  most  brilliant  productions  that  have  made  their  appearance  in 
recent  decades. 

DANIELS'S   ELEMENTS  OF  PUBUC  FINANCE 

By  WiNTHROP  More  Daniels,  Professor  of  Political  Econ- 
omy in  Princeton  University.    373  pp.     12mo.     $1.50. 

r,  Spencer  Baldwin,  Professor  in  Boston  University ;— It  is  a  piece  of 
work  well  done  both  from  a  scientific  and  a  literary  point  of  view— a  text-book 
with  a  style.  .  .  .  The  lucid  explanation  of  the  financial  system  of  the 
United  States  makes  the  book  particularly  valuable  for  the  American  studeat. 

SCOTT'S   MONEY   AND  BANKING 

By  "W.  A.  Scott,  Professor  in  the  University  of  Wisconsin. 
381  pp.     8vo.     $3.00. 

H.  E.  Mills,  Professor  in  Vassar  College :— It  is  clear,  comprehensive,  and 
conservative.  All  in  all,  it  seems  to  me  the  best  single  book  to  use  in  connec- 
tion with  a  course  on  Money  and  Banking. 

SEAGER'S  INTRODUCTION  TO  ECONOMICS 

By  Henry  R.  Seager,  Professor  in  Columbia  University. 
Third  Edition,  Revised  and  Enlarged.     604  pp.     8vo.    $2.25. 

Journal  of  Political  Economy :— Thoroughly  modem  in  doctrine  ;  wide  in 
sympathy,  clear,  sprightly,  and  stimulating  in  style  and  manner  of  presentation. 

SEAGER'S  ECONOMICS,  BRIEFER  COURSE 

By  Henry  R.  Seager,  Professor  in  Columbia  University. 
467  pp.     Large  12mo. 

Intended  primarily  for  those  who  wish  to  give  only  that  amount 
of  attention  to  economic  theory  that  is  essential  to  the  intelli- 
gent discussion  of  practical  economic  problems. 


HENRY    HOLT    AND     COMPANY 

34  West  33d  Street  New  York 


BY   THE   LATE    FRANCIS   A.   WALKER 

DISCUSSIONS  IN  ECONOMICS  AND  STATISTICS 

Edited  by  Professor  Davis  R.  Dewey.  With  portrait.  2  vols. 
8vo.    454  +  481  pp.     $6.00  net. 

Important  papers  by  a  great  authority  on  Finance,  Taxation, 
Money,  Bimetallism,  Economic  Theory,  Statistics,  National 
Growth,  Social  Economics,  etc. 

The  Dial :— .  .  .  Economics  in  the  hands  of  this  master  was  no  diamai 
science,  because  of  his  broad  sympathies,  his  healthy,  conservative  optimism, 
his  belief  in  the  efficacy  of  efEort ;  and,  in  a  more  superficial  sense,  because  of 
his  saving  sense  of  humor  and  his  happy  way  of  putting  things  ....  he 
was  the  fortunate  possessor  of  a  very  pleasing  literary  style  .  .  .  .  clear  and 
interesting  to  the  general  reader,  as  well  as  instructive  to  the  careful  student. 
There  could  have  been  no  more  fitting  monument  to  his  memory  than  these 
two  volumes,  together  with  the  other  volume  of  "  Discussions  in  Education." 

MONEY 

550  pp.     12mo.     $2.00. 

New  York  Tribune  :— The  essential  facts  of  monetary  experience  in  every 
country  are  presented  with  sufficient  fullness  and  with  judicious  mingling  of 
authority  on  disputed  points.  The  work  will  win  a  very  honornble  place  for 
its  author  among  the  few  who  are  advancing  toward  the  mastery  of  a  most  dif- 
ficult science. 

MONEY  IN  ITS  RELATIONS  TO  TRADE  AND  INDUSTRY 

339  pp.     12mo.     $1.25. 

Boston  Courier :— The  present  volume  is  of  a  more  popular  natore  than  his 
previous  one  on  Money,  but  certainly  is  not  on  that  account  less  important. 
Viewed  in  its  immediate  relation  to  the  money  questions  of  the  day  which  are 
entering  more  and  more  into  politics  and  becoming  therefore  active  levers  for 
the  advancement  of  society  ....  adapted  to  easy  comprehension  by  a 
mixed  audience,  it  is  a  publication  of  greater  moment  than  its  more  elaborate 
and  critical  predecessor. 

INTERNATIONAL   BIMETALUSM 

297  pp.     12mo.     $1.25. 

The  Outlook :— The  best  book  yet  published  in  the  English  language  for  the 
^Tposition  of  the  distinctively  economic  questions  at  issue  between  bimetallistc 
azA  monometallists. 

WAGES 

A  Treatise  on  Wages  and  the  Wages  Class,  428  pp.  12mo. 
$2.00. 

Nation:— The  most  complete  and  exhaustive  treatise  on  the  wages  question 
with  which  we  are  acquainted.  .  .  .  The  general  correctness  of  its  line  of 
argument  is  in  striking  contrast  to  much  that  has  been  written  on  the  subject. 

POLITICAL  ECONOMY 

Advanced  Course.  537  pp.  8vo.  $2.00  net. 
Briefer  Course.  415  pp.  12mo.  $1.20  net. 
Elementary  Course.  323  pp.     12mo.   $1.00  net. 

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american  ipublic  problems  Series 

Edited  by  Ralph  Curtis  Ringwalt 

Chinese  Immigration 

By  Mary  Roberts  Coolidge,  Formerly  Associate  Professor 
of  Sociology  in  Stanford  University.  531  pp.,  $1.75  net;  by 
mail,  $1.90.     {Just  issued.) 

Preseiats  the  most  comprehensive  record  of  the  Chinaman  in 
the  United  States  that  has  yet  been  attempted. 

"Scholarly.     Covers  every  important  phase,  economic,  social,  and 

Eolitical,  of  the  Chinese  question  in  America  down  to  the  San  Francisco 
re  in  1906." — New  York  Sun. 

"  Statesmanlike.    Of  intense  interest." — Hartford  Courant. 
"  A  remarkably  thorough  historical  study.    Timely  and  useful.    En- 
hanced by  the  abundant  array  of  documentary  facts  and  evidence." — 
Chicago  Record- Herald. 

Immigration:  And   Its  Effects   Upon  the  United 
States 

By  Prescott  F.  Hall,  A.B.,  LL.B,  Secretary  of  the  Immi- 
gration Restriction  League.    393  pp.   $1.50  net;  by  mail,  $1.65. 

"  Should  prove  interesting;  to  everyone.  Very  readable,  forceful  and 
convincing.  Mr.  Hall  considers  every  possible  phase  of  this  great 
question  and  does  it  in  a  masterly  way  that  shows  not  only  that  he 
thoroughly  understands  it,  but  that  he  is  deeply  interested  in  it  and  has 
studied  everything  bearing  upon  it." — Boston  Transcript. 

"A  readable  work  containing  a  vast  amount  of  valuable  information. 
Especially  to  be  commended  is  the  discussion  of  the  racial  effects.  As  a 
trustworthy  general  guide  it  should  prove  a  god-send." — New  York 
Evening  Post. 

The  Election  of  Senators 

By  Professor  George  H.  Haynes,  Author  of  "  Representation 
in  State  Legislatures."     300  pp.    $1.50  net;  by  mail,  $1.65. 

Shows   the  historical  reasons  for  the  present  method,  and 

its  effect  on  the  Senate  and  Senators,  and  on  state  and  local 

government,  with  a  detailed  review  of  the  arguments  for  and 

against  direct  election. 

"  A  timely  book.  .  .  .  Prof.  Haynes  is  qualified  for  a  historical  and 
analytical  treatise  on  the  subject  of  the  Senate."— .A^^w  York  Evening  Sun, 

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NEW  BOOKS  ON  THE  LIVING  ISSUES  BY  LIVING 
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Cloth  Bound  50c  per  volume  net ;  by  mail  56c. 

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Every  volume  is  absolutely  new,  and  specially  written  for 
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Every  volume  is  sold  separately.  Each  has  illustrations 
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Every  volume  is  written  by  a  recognized  authority  on  its 
subject,  and  the  Library  is  published  under  the  direction  of 
four  eminent  Anglo-Saxon  scholars  —  Gilbert  Murray,  of 
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of  Aberdeen;  and  Prof.  W.  T.  Brewster,  of  Columbia. 

Every  subject  is  of  living  and  permanent  interest.  These 
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Each  volume  is  complete  and  independent ;  but  the  series 
has  been  carefully  planned  as  a  whole  to  form  a  compre- 
hensive library  of  modern  knowledge  covering  the  chief  sub- 
jects in  History  and  Geograph}^  Literature  and  Art,  Science, 
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SOME  COMMENTS  ON  THE  SERIES  AS  A   WHOLE: 

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"The    excellence   of   these   books." — The   Dial. 

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VOLUMES  ON  SOCIAL  SCIENCE  NOW  READY 

Ethics  —  By  G.  E.  Moore.  The  School 
Missions  By  J.   J.   FiNDLAY. 

By   Mrs.    Creighton.  The  Stock  Exchange 
The  Elements   of   Political  By  F.   W.  Hirst. 

Economy  Parliament 

By  S.  J.  Chapman.  By  C.   P.  Ilbert. 

The  Socialist  Movement  The  Evolution  of  Industry 

By  J.  R.  Macdonald.  By  D.  H.  Macgregor. 

The  Science  of  Wealth  Elements  of  English  Law 

By  J.  A.  HoBSON.  By  W.  M.  Geldart. 


HENRY     HOLT     AND     COMPANY 

34  WEST  33rd  ST.  (VII1'12)  NEW  YORK 


ORTH'S  SOCIALISM  AND  DEMOCRACY  IN  EUROPE 

By  Samuel  P.  Orth,  Author  of  "  Five  American  Politicians," 
"  Centralization  of  Administration  in  Ohio,"  etc.  $1.50  net; 
by  mail,  11.62. 

Traces  briefly  the  spread  of  the  Socialist  movement  in  France,  Belgium, 
Germany,  and  England,  and  attempts  to  determine  the  relation  of  economic 
and  political  Socialism  to  democracy — a  question  of  peculiar  interest  to  the 
friends  of  the  American  Republic  at  this  time.  The  author  has  made  ex- 
tended visits  to  the  countries  studied.  He  has  tried  to  catch  the  spirit  of  the 
movement  by  personal  contact  with  the  Socialist  leaders  and  their  antagonists, 
and  by  manr  interviews  with  laboring  men,  the  rank  and  file  in  every  country 
visited. 

The  contents  include  :  The  Development  of  Socialism— The  Political 
Awakening  of  Socialism  :  The  Period  of  Revolution— The  Political  Awakening 
of  Socialism:  The  International— The  Socialist  Party  of  France — The  Belgian 
Labor  Party — The  German  Social-Democracy — The  English  Labor  Party — 
Conclusion— .\  Very  Full  Appendix,  including  a  Bibliography,  "  Programs  " 
of  Socialists  in  different  countries,  etc. 

"  A  condensed  study  of  the  history  of  Socialism  and  of  the  present  status  of 
the  movement  in  the  countries  where  it  has  made  the  most  progress.  He  writes 
as  a  sympathetic  student  rather  than  as  a  soc[2L\ist."—Spring/ield  Republican. 

SIMKHOVITCH'S  MARXISM  VERSUS  SOCIALISM 

By  V.  G.  SiMKHOViTCH,  Associate  Professor  of  Economic 
History  Columbia  University.   12mo.   Probable  price,  $1. 75  net. 

Professor  Simkhovitch's  work,  the  result  of  many  years  of  study,  furnishes 
a  thorough  and  intimate  knowledge  of  all  the  intricate  theories,  problems  and 
difficulties  of  modern  socialism.  Marx's  Socialism  is  based  on  his  interpre- 
tation of  economic  history  and  economic  tendencies.  According  to  Marx, 
these  tendencies  make  Socialism  inevitable.  Were  economic  conditions  and 
tendencies  different.  Socialism  would  have  been  impossible.  Professor 
Simkhovitch  shows  us  that  the  economic  tendencies  of  to-day  are  quite  differ- 
ent from  what  Marx  expected  them  to  be  and  that  socialism  from  the  stand- 
point of  Marx's  own  theory  it  quite  impossible.  Marxism  had  thus  turned 
against  socialism,  and  the  revisionist,  the  reformist,  the  Back-to-Kant,  the 
syndicalist  and  other  movements  represent  a  quest  for  a  possible  new  mean- 
ing for  the  word. 


HENRY     HOLT     AND     COMPANY 

PUBLISHERS  in  '13  NEW   YORK 


UNIVERSITY  OF  CALIFORNIA  AT  UOS  ANGEUES 

THE  UNIVERSITY  LIBRARY 

This  "book  is  DUE  on  the  last  date  stamped  below 


MOV  12  1W7 

APP.4    1951 
NOV  2  9^^^^ 

OCT  2-  1958 

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